Palindromic Quasi_Pronics of the form n(n+X)

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Palindromic Quasipronics of the form n(n+x)
n(n+0) n(n+1) n(n+2) n^2+1 n^2+x n^2–x n^2+(n+1) n^2+(n+x)

Introduction

Palindromic numbers are numbers which read the same from

left to right (forwards) as from the right to left (backwards)

Here are a few random examples : 7, 3113, 44611644

Quasi_Pronic numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base ( base + X )

PLAIN TEXT CONSEMOR

Palindromic Quasipronics

Quasipronics of the form x * (x + 0) can only end with a 0, 1, 4, 5, 6 or 9.
And if it terminates in 5, it terminates in 25 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

1 can only be followed by an even number : 10, 12, 14, 16 or 18

4 can only be followed by an even number : 40, 42, 44, 46 or 48

5 can only be followed by 2 : 52

6 can only be followed by an odd number : 61, 63, 65, 67 or 69

9 can only be followed by an even number : 90, 92, 94, 96 or 98

Also

The sum of the first n odd numbers is palindromic.

Sequence = 1 + 3 + 5 + 7 + 9 + 11 + ... + z

Main source see Palindromic Square Numbers

Index entries in OEIS A000290, A002778 and A002779

Quasipronics of the form x * (x + 1) can only end with a 0, 2 or 6. And if it terminates in 6, it terminates in 06 or 56 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

2 can be followed by any number : 20, 21, 22, 23, 24, 25, 26, 27, 28 or 29

6 can only be followed by 0 or 5 : 60 or 65

Also

The sum of the first n even numbers is palindromic.

Sequence = 2 + 4 + 6 + 8 + 10 + 12 + ... + z

Main source see Palindromic Pronic Numbers

Index entries in OEIS A002378, A028336 and A028337

Quasipronics of the form x * (x + 2) can only end with a 0, 3, 4, 5, 8 or 9.
And if it terminates in 4, it terminates in 24; if it terminates in 9, it terminates in 99 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

Main source see Palindromic QuasiPronic Numbers

3 can only be followed by an even number : 30, 32, 34, 36 or 38

4 can only be followed by 2 : 42

5 can only be followed by an odd number : 51, 53, 55, 57 or 59

8 can only be followed by an even number : 80, 82, 84, 86 or 88

9 can only be followed by 9 : 99

Quasipronics of the form x * (x + 3) can only end with a 0, 4 or 8. And if it terminates in 4, it terminates in 04 or 54 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

4 can only be followed by 0 or 5 : 40 or 45

8 can be followed by any number : 80, 81, 82, 83, 84, 85, 86, 87, 88 or 89

There are no palindromic quasipronics of type x * (x + 3) of lengths 8, 16, 18, 20, 24, 26, 28, 36, 42, 48, 50, 54.

Also

The palindromic sums of the first n even composites. See my webpage firstpal.htm#sfn5.

Sequence = 4 + 6 + 8 + 10 + 12 + 14 + ... + z

Index entries in OEIS A028553, A058851 and A028554

E.g. : 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = [9² + 9 – 2] = n² + n – 2 = 88 = [8 * (8 + 3)] = x * (x + 3)

The following substitution shows why the sequence equals x * (x + 3)

The sequence can be rewritten as 2 * (2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ...)
The second term is 'the sum of the natural numbers' minus the starting number 1.
Replace it with the general formula we get 2 * [ (n^2 + n)/2 – 1 ] or [ n^2 + n – 2 ]
Substitute n with (x + 1) we get [ (x + 1)^2 + (x + 1) – 2 ] and work it out.
The sequence evolves from [ x^2 + 2x + 1 + x + 1 – 2 ] to [ x^2 + 3x ]
which finally yields to x * (x + 3) [ QED ]

Quasipronics of the form x * (x + 4) can only end with a 0, 1, 2, 5, 6 or 7.
And if it terminates in 1, it terminates in 21; if it terminates in 6, it terminates in 96 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

1 can only be followed by 2 : 12

2 can only be followed by an odd number : 21, 23, 25, 27 or 29

5 can only be followed by an even number : 50, 52, 54, 56 or 58

6 can only be followed by 9 : 69

7 can only be followed by an odd number : 71, 73, 75, 77 or 79

There are no palindromic quasipronics of type x * (x + 4) of lengths 12, 22, 24, 42, 46.

Also

The palindromes of the form n² – 4 whereby x equals n – 2.

Proof by substitution :

x * ( x + 4 )
(n – 2) * ( (n – 2) + 4 )
(n – 2) * (n + 2)
n² + 2n –2n – 4
n² – 4 [ QED ]

Quasipronics of the form x * (x + 5) can only end with a 0, 4 or 6.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

4 can be followed by any number : 40, 41, 42, 43, 44, 45, 46, 47, 48 or 49

6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69

There are no palindromic quasipronics of type x * (x + 5) of lengths 4, 8, 12, 15, 18, 19, 23, 24, 28, 32, 36, 42, 52, 53.

Quasipronics of the form x * (x + 6) can only end with a 0, 1, 2, 5, 6 or 7.
And if it terminates in 1, it terminates in 91; if it terminates in 6, it terminates in 16 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

1 can only be followed by 9 : 19

2 can only be followed by an odd number : 21, 23, 25, 27 or 29

5 can only be followed by an odd number : 51, 53, 55, 57 or 59

6 can only be followed by 1 : 61

7 can only be followed by an even number : 70, 72, 74, 76 or 78

There are no palindromic quasipronics of type x * (x + 6) of lengths 4, 9, 10, 11, 26, 34, 45, 46, 53.

Quasipronics of the form x * (x + 7) can only end with a 0, 4 or 8. And if it terminates in 4, it terminates in 44 or 94 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

4 can only be followed by 4 or 9 : 44 or 49

8 can be followed by any number : 80, 81, 82, 83, 84, 85, 86, 87, 88 or 89

There are no palindromic quasipronics of type x * (x + 7) of lengths 4.

Quasipronics of the form x * (x + 8) can only end with a 0, 3, 4, 5, 8 or 9.
And if it terminates in 4, it terminates in 84; if it terminates in 9, it terminates in 09 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

3 can only be followed by an odd number : 31, 33, 35, 37 or 39 { This 3956... produces ever expanding Mandelbrot-like infinite palindromes from length 37 on }.

4 can only be followed by 8 : 48

5 can only be followed by an even number : 50, 52, 54, 56 or 58

8 can only be followed by an even number : 80, 82, 84, 86 or 88

9 can only be followed by 0 : 90

There are no palindromic quasipronics of type x * (x + 8) of lengths 3, 50.

Quasipronics of the form x * (x + 9) can only end with a 0, 2 or 6. And if it terminates in 6, it terminates in 36 or 86 only.
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

2 can be followed by any number : 20, 21, 22, 23, 24, 25, 26, 27, 28 or 29

6 can only be followed by 1 or 8 : 63 or 68

There are no palindromic quasipronics of type x * (x + 9) of lengths 6, 10, 14, 20, 32, 36, 38, 54, 55.

Case Quapron	Change of variables	CUDApalin parameters	Base Correction
x(x+3)	n.a.	A B C → 1 3 0	base = CUDAbase
x(x+4)	n.a.	A B C → 1 4 0	base = CUDAbase
x(x+5)	n.a.	A B C → 1 5 0	base = CUDAbase
x(x+6)	n.a.	A B C → 1 6 0	base = CUDAbase
x(x+7)	n.a.	A B C → 1 7 0	base = CUDAbase
x(x+8)	n.a.	A B C → 1 8 0	base = CUDAbase
x(x+9)	n.a.	A B C → 1 9 0	base = CUDAbase

An interesting infinite palindromic pattern hides in the list for case X = 3

28 x 31 = 868
298 x 301 = 89698
2998 x 3001 = 8996998
29998 x 30001 = 899969998
299998 x 300001 = 89999699998
2999998 x 3000001 = 8999996999998
29999998 x 30000001 = 899999969999998

Two intertwined and nice, alas finite palindromic patterns hide in the list for case X = 4

Here are the two basenumbers
14{5544}_n for n=0,1,2,3,4
and
14{5544}52_n for n=0,1,2,3,4

A very nice but finite palindromic pattern hides in the list for case X = 5

202 x 207 = 41814
20402 x 20407 = 416343614
2040402 x 2040407 = 4163250523614

A very nice infinite twofold palindromic pattern hides in the list for case X = 7

19 x 26 = 494
219 x 226 = 49494
2109 x 2116 = 4462644
21009 x 21016 = 441525144
210009 x 210016 = 44105250144
2100009 x 2100016 = 4410052500144
21000009 x 21000016 = 441000525000144

Note that if you add n to n+7 and subtract 1 from the above pattern
you get also a palindromic sequence !

19 + 26 – 1 = 44
219 + 226 – 1 = 444
2109 + 2116 – 1 = 4224
21009 + 21016 – 1 = 42024
210009 + 210016 – 1 = 420024
2100009 + 2100016 – 1 = 4200024
21000009 + 21000016 – 1 = 42000024

And then there is this astonishing infinite pattern where the multiplier n+7 is palindromic itself !

902 x 909 = 819918
9002 x 9009 = 81099018
90002 x 90009 = 8100990018
900002 x 900009 = 810009900018
90000002 x 90000009 = 81000099000018

Embedded palindromes can be found in some multiplicands starting with 20 and ending with 004

[85] 20 99009888890099 004
[86] 20 98890099009889 004
[91] 20 990098888890099 004

Another beautiful infinite palindromic pattern hides in the list for case X = 8

66 x 74 = 4884
696 x 704 = 489984
6996 x 7004 = 48999984
69996 x 70004 = 4899999984
699996 x 700004 = 489999999984
6999996 x 7000004 = 48999999999984

Admire this surprising base duet in the list for case X = 9

1544465454355544 [72]
1544465454355544 [73]

It is just a matter of changing two fours into fives and two fives into fours !

Can you detect more hidden patterns in these sequences ?

The Table

Index Nr	Basenumber	Length
Index Nr	Palindromic Quasipronics of form n(n+X)	Length

	[PG4] Case X = 0 [Scanned exhaustively up to length 67]
See in-depth webpage about Palindromic Square Numbers

	[QP1] Case X = 1 [Scanned exhaustively up to length 60]
See in-depth webpage about Palindromic Pronic Numbers

	[QP2] Case X = 2 [Scanned exhaustively up to length 57]
See in-depth webpage about Palindromic Quasipronic Numbers

	[QP3] Case X = 3
One can find the regular numbers of the form n(n+3) at %N n(n+3). under A028552. The palindromic numbers of the form n(n+3) are categorised as follows : %N n(n+3) is a palindrome. under A028553. %N Palindromes of the form n(n+3). under A028554. More terms from Robert Xiao, Nov 11, 2022. See Index Nrs 72 up to 77. More terms from Robert Xiao, Dec 6, 2022. See Index Nrs 78 up to 92. More terms from Patrick De Geest, Jun 18, 2023. See Index Nr 93. More terms from Patrick De Geest, Jul 12, 2023. See Index Nrs 94 up to 96.
96	2.999.999.999.999.999.999.999.999.998	28
96	8.999.999.999.999.999.999.999.999.996.999.999.999.999.999.999.999.999.998	55
95	2.988.775.376.311.964.100.375.012.399	28
95	8.932.778.250.048.722.618.890.980.456.540.890.988.162.278.400.528.772.398	55
94	2.138.799.321.241.622.416.594.967.171	28
94	4.574.462.536.543.624.762.161.771.233.321.771.612.674.263.456.352.644.754	55
93	299.999.999.999.999.999.999.999.998	27
93	89.999.999.999.999.999.999.999.999.699.999.999.999.999.999.999.999.998	53
92	67.266.752.862.975.704.263.041.726	26
92	4.524.816.040.728.650.106.101.807.447.081.016.010.568.270.406.184.254	52
91	299.99.999.999.999.999.999.999.998	26
91	899.999.999.999.999.999.999.999.969.999.999.999.999.999.999.999.998	51
90	29.992.506.467.046.072.308.055.999	26
90	899.550.444.175.800.470.083.640.545.046.380.074.008.571.444.055.998	51
89	29.447.193.289.689.589.756.536.744	26
89	867.137.192.640.339.603.223.179.525.971.322.306.933.046.291.731.768	51
88	29.217.285.176.188.752.980.568.134	26
88	853.649.753.066.739.050.298.194.030.491.892.050.937.660.357.946.358	51
87	20.222.375.405.486.915.221.395.156	26
87	408.944.467.040.442.078.820.160.212.061.028.870.244.040.764.449.804	51
86	20.126.438.211.248.062.653.407.896	26
86	405.073.515.071.186.115.853.603.050.306.358.511.681.170.515.370.504	51
85	2.999.999.999.999.999.999.999.998	25
85	8.999.999.999.999.999.999.999.996.999.999.999.999.999.999.999.998	49
84	2.982.685.662.335.431.618.868.233	25
84	8.896.413.760.301.352.404.471.977.791.744.042.531.030.673.146.988	49
83	299.999.999.999.999.999.999.998	24
83	89.999.999.999.999.999.999.999.699.999.999.999.999.999.999.998	47
82	298.345.284.058.522.231.035.523	24
82	89.009.908.519.960.319.280.639.193.608.291.306.991.580.990.098	47
81	67.121.888.369.233.446.819.631	23
81	4.505.347.898.251.836.262.895.995.982.626.381.528.987.435.054	46
80	29.999.999.999.999.999.999.998	23
80	899.999.999.999.999.999.999.969.999.999.999.999.999.999.998	45
79	20.080.254.557.051.380.400.781	23
79	403.216.623.075.982.729.302.909.203.927.289.570.326.612.304	45
78	20.077.966.701.734.324.735.381	23
78	403.124.746.875.952.318.571.000.175.813.259.578.647.421.304	45
77	9.177.267.147.903.696.840.248	22
77	84.222.232.303.992.454.255.555.555.245.429.930.323.222.248	44
76	6.764.462.802.994.617.254.971	22
76	45.757.957.013.097.794.051.966.915.049.779.031.075.975.754	44
75	6.723.378.107.590.656.205.551	22
75	45.203.813.177.629.313.451.699.615.431.392.677.131.830.254	44
74	2.999.999.999.999.999.999.998	22
74	8.999.999.999.999.999.999.996.999.999.999.999.999.999.998	43
73	299.999.999.999.999.999.998	21
73	89.999.999.999.999.999.999.699.999.999.999.999.999.998	41
72	92.882.981.610.737.690.369	20
72	8.627.248.272.900.635.954.334.595.360.092.728.427.268	40
71	29.999.999.999.999.999.998	20
71	899.999.999.999.999.999.969.999.999.999.999.999.998	39
70	29.848.768.829.132.626.523	20
70	890.949.000.614.999.508.181.805.999.416.000.949.098	39
69	9.382.541.482.587.317.008	19
69	88.032.084.672.471.808.733.780.817.427.648.023.088	38
68	2.999.999.999.999.999.998	19
68	8.999.999.999.999.999.996.999.999.999.999.999.998	37
67	Prime! 2.915.956.745.919.295.163	19
67	8.502.803.744.072.244.896.984.422.704.473.082.058	37
66	2.124.902.219.343.678.986	19
66	4.515.209.441.771.692.447.442.961.771.449.025.154	37
65	299.999.999.999.999.998	18
65	89.999.999.999.999.999.699.999.999.999.999.998	35
64	201.273.817.252.896.176	18
64	40.511.149.511.552.246.964.225.511.594.111.504	35
63	200.007.029.236.835.996	18
63	40.002.811.744.144.569.496.544.144.711.820.004	35
62	63.981.601.780.397.661	17
62	4.093.645.366.385.385.335.835.836.635.463.904	34
61	29.999.999.999.999.998	17
61	899.999.999.999.999.969.999.999.999.999.998	33
60	29.047.072.823.714.849	17
60	843.732.439.626.193.818.391.626.934.237.348	33
59	20.094.657.512.269.781	17
59	403.795.260.535.420.404.024.535.062.597.304	33
58	9.446.469.351.863.299	16
58	89.235.783.215.692.644.629.651.238.753.298	32
57	9.130.277.032.071.258	16
57	83.361.958.682.367.966.976.328.685.916.338	32
56	6.715.682.777.346.386	16
56	45.100.395.165.946.888.864.956.159.300.154	32
55	2.999.999.999.999.998	16
55	8.999.999.999.999.996.999.999.999.999.998	31
54	936.880.964.592.243	15
54	877.745.941.815.294.492.518.149.547.778	30
53	936.259.984.512.718	15
53	876.582.758.599.757.757.995.857.285.678	30
52	918.039.000.965.248	15
52	842.795.607.293.273.372.392.706.597.248	30
51	910.127.769.311.068	15
51	828.332.556.471.143.341.174.655.233.828	30
50	299.999.999.999.998	15
50	89.999.999.999.999.699.999.999.999.998	29
49	212.353.038.141.391	15
49	45.093.812.807.879.697.870.821.839.054	29
48	29.999.999.999.998	14
48	899.999.999.999.969.999.999.999.998	27
47	29.854.146.767.798	14
47	891.270.079.233.313.332.970.072.198	27
46	28.814.579.197.814	14
46	830.279.974.347.181.743.479.972.038	27
45	2.999.999.999.998	13
45	8.999.999.999.996.999.999.999.998	25
44	2.013.969.414.821	13
44	4.056.072.803.840.483.082.706.504	25
43	2.010.463.667.561	13
43	4.041.964.158.588.858.514.691.404	25
42	299.999.999.998	12
42	89.999.999.999.699.999.999.998	23
41	287.446.679.729	12
41	82.625.593.688.088.639.552.628	23
40	283.162.754.884	12
40	80.181.145.754.345.754.118.108	23
39	202.055.469.356	12
39	40.826.412.697.279.621.462.804	23
38	200.952.736.816	12
38	40.382.002.434.443.420.028.304	23
37	91.222.338.358	11
37	8.321.515.015.775.105.151.238	22
36	29.999.999.998	11
36	899.999.999.969.999.999.998	21
35	21.280.530.846	11
35	452.860.993.151.399.068.254	21
34	2.999.999.998	10
34	8.999.999.996.999.999.998	19
33	2.946.920.844	10
33	8.684.342.469.642.434.868	19
32	2.134.473.706	10
32	4.555.978.008.008.795.554	19
31	299.999.998	9
31	89.999.999.699.999.998	17
30	294.174.669	9
30	86.538.736.763.783.568	17
29	294.127.328	9
29	86.510.885.958.801.568	17
28	289.053.683	9
28	83.552.032.523.025.538	17
27	212.325.206	9
27	45.081.993.739.918.054	17
26	29.999.998	8
26	899.999.969.999.998	15
25	28.862.883	8
25	833.066.101.660.338	15
24	21.341.691	8
24	455.467.838.764.554	15
23	9.443.423	7
23	89.178.266.287.198	14
22	2.999.998	7
22	8.999.996.999.998	13
21	2.861.419	7
21	8.187.727.277.818	13
20	2.122.206	7
20	4.503.764.673.054	13
19	636.776	6
19	405.485.584.504	12
18	299.998	6
18	89.999.699.998	11
17	Prime! 212.671	6
17	45.229.592.254	11
16	212.206	6
16	45.032.023.054	11
15	63.701	5
15	4.058.008.504	10
14	29.998	5
14	899.969.998	9
13	29.369	5
13	862.626.268	9
12	28.814	5
12	830.333.038	9
11	2.998	4
11	8.996.998	7
10	2.126	4
10	4.526.254	7
9	671	3
9	452.254	6
8	298	3
8	89.698	5
7	Prime! 211	3
7	45.154	5
6	88	2
6	8.008	4
5	66	2
5	4.554	4
4	28	2
4	868	3
3	8	1
3	88	2
2	1	1
2	4	1
1	0	1
1	0	1

	[QP4] Case X = 4
One can find the regular numbers of the form n(n+4) at %N n(n+4). [= m^2 – 4] under A028347. The palindromic numbers of the form n(n+4) are categorised as follows : %N n(n+4) is a palindrome. under A028555. %N Palindromes of the form n(n+4). under A028556. More terms from Robert Xiao, Nov 11, 2022. See Index Nrs 86 up to 97. More terms from Robert Xiao, Dec 6, 2022. See Index Nrs 98 up to 114. More terms from Patrick De Geest, Jun 18, 2023. See Index Nrs 115 up to 119. More terms from Patrick De Geest, Jun 12, 2023. See Index Nrs 120 up to 124.
124	1.449.997.977.750.905.484.114.663.494	28
124	2.102.494.135.481.715.395.332.794.456.544.972.335.935.171.845.314.942.012	55
123	1.098.555.144.306.708.368.115.652.603	28
123	1.206.823.405.082.732.847.096.661.535.351.666.907.482.372.805.043.286.021	55
122	710.571.130.491.555.462.106.283.845	27
122	504.911.331.488.047.140.574.695.358.853.596.475.041.740.884.133.119.405	54
121	520.776.512.106.154.851.929.399.222	27
121	271.208.175.561.452.051.050.606.848.848.606.050.150.254.165.571.802.172	54
120	462.501.098.643.281.865.730.935.852	27
120	213.907.266.246.242.742.861.904.368.863.409.168.247.242.642.662.709.312	54
119	263.334.882.575.895.582.010.126.118	27
119	69.345.260.381.260.714.597.571.490.609.417.579.541.706.218.306.254.396	53
118	242.591.662.097.400.206.120.517.165	27
118	58.850.714.519.179.199.773.324.463.836.442.337.799.197.191.541.705.885	53
117	224.704.366.117.106.965.856.830.095	27
117	50.492.052.152.090.849.047.799.348.884.399.774.094.809.025.125.029.405	53
116	159.205.876.240.607.402.599.330.064	27
116	25.346.511.029.539.600.663.713.962.526.931.736.600.693.592.011.564.352	53
115	145.149.880.278.760.511.394.779.502	27
115	21.068.487.744.938.509.633.089.177.977.198.033.690.583.944.778.486.012	53
114	34.727.644.457.529.973.575.184.603	26
114	1.206.009.289.568.612.292.630.513.113.150.362.922.168.659.829.006.021	52
113	14.644.141.038.967.078.813.101.802	26
113	214.450.866.769.159.794.512.784.161.487.215.497.951.967.668.054.412	51
112	11.060.878.084.645.885.718.171.113	26
112	122.343.024.003.399.637.426.192.141.291.624.736.993.300.420.343.221	51
111	10.999.718.519.705.378.502.881.093	26
111	120.993.807.512.749.483.323.588.909.885.323.384.947.215.708.399.021	51
110	10.988.066.442.555.347.757.935.293	26
110	120.737.604.146.010.935.489.525.919.525.984.539.010.641.406.737.021	51
109	4.602.538.038.508.234.235.379.452	25
109	21.183.356.395.915.224.245.353.866.835.354.242.251.959.365.338.112	50
108	2.287.078.595.197.804.413.027.571	25
108	5.230.728.500.611.962.503.102.929.292.013.052.691.160.058.270.325	49
107	2.236.630.247.653.903.734.348.251	25
107	5.002.514.864.720.362.751.166.701.076.611.572.630.274.684.152.005	49
106	Prime! 354.965.777.552.293.665.391.223	24
106	126.000.703.233.304.429.440.576.675.044.924.403.332.307.000.621	48
105	230.086.670.895.165.888.115.825	24
105	52.939.876.123.620.377.389.648.384.698.377.302.632.167.893.925	47
104	165.463.784.922.972.904.635.874	24
104	27.378.264.121.035.835.512.925.352.921.553.853.012.146.287.372	47
103	112.472.334.234.620.030.086.323	24
103	12.650.025.968.184.080.828.964.946.982.808.048.186.952.005.621	47
102	Prime! 110.758.273.061.497.927.187.533	24
102	12.267.395.051.565.337.420.524.242.502.473.356.515.059.376.221	47
101	109.937.743.582.989.132.653.443	24
101	12.086.307.464.119.068.214.771.017.741.286.091.146.470.368.021	47
100	109.881.103.994.599.780.425.293	24
100	12.073.857.015.072.051.822.607.170.622.815.027.051.075.837.021	47
99	28.125.704.708.114.962.863.849	23
99	791.055.265.328.080.188.386.131.683.881.080.823.562.550.197	45
98	26.456.664.770.722.522.496.598	23
98	699.955.110.790.390.223.863.181.368.322.093.097.011.559.996	45
97	8.671.150.261.966.794.104.367	22
97	75.188.846.865.606.802.022.799.722.020.860.656.864.888.157	44
96	8.669.770.269.474.237.754.629	22
96	75.164.916.525.459.397.132.299.223.179.395.452.561.946.157	44
95	7.381.657.651.450.977.180.605	22
95	54.488.869.683.224.756.112.511.521.165.742.238.696.888.445	44
94	2.645.642.130.670.011.258.498	22
94	6.999.422.283.576.156.926.572.756.296.516.753.822.249.996	43
93	2.242.706.052.383.514.679.451	22
93	5.029.730.437.397.648.089.427.249.808.467.937.340.379.205	43
92	278.687.223.973.055.857.539	21
92	77.666.568.805.808.199.470.807.499.180.850.886.566.677	41
91	262.806.806.089.087.049.308	21
91	69.067.417.326.747.001.778.187.710.074.762.371.476.096	41
90	109.561.045.720.804.636.993	21
90	12.003.622.739.436.244.059.595.044.263.493.722.630.021	41
89	88.977.974.712.438.626.397	20
89	7.917.079.983.927.367.660.220.667.637.293.899.707.197	40
88	88.971.166.676.682.068.647	20
88	7.915.868.499.809.941.775.885.771.499.089.948.685.197	40
87	88.005.752.595.853.210.939	20
87	7.745.012.489.962.524.175.995.714.252.699.842.105.477	40
86	50.696.930.271.124.020.264	20
86	2.570.178.738.915.211.026.996.201.125.198.378.710.752	40
85	14.554.455.445.544.554.452	20
85	211.832.173.316.341.535.101.535.143.613.371.238.112	39
84	8.582.154.816.793.069.369	19
84	73.653.381.299.404.482.100.128.440.499.218.335.637	38
83	8.337.580.895.150.012.458	19
83	69.515.255.183.170.483.066.038.407.138.155.251.596	38
82	2.782.718.512.264.031.307	19
82	7.743.522.318.496.943.766.673.496.948.132.253.477	37
81	Prime! 749.333.605.100.067.511	18
81	561.500.851.732.263.925.529.362.237.158.005.165	36
80	710.620.641.519.688.845	18
80	504.981.696.153.854.124.421.458.351.696.189.405	36
79	501.710.484.827.137.032	18
79	251.713.410.585.480.900.009.084.585.014.317.152	36
78	282.062.766.593.827.799	18
78	79.559.404.298.564.180.508.146.589.240.495.597	35
77	229.161.536.782.656.475	18
77	52.515.009.940.588.817.471.888.504.990.051.525	35
76	145.544.554.455.445.544	18
76	21.183.217.331.634.153.935.143.613.371.238.112	35
75	85.912.342.960.955.027	17
75	7.380.930.673.040.759.119.570.403.760.390.837	34
74	28.172.587.468.596.949	17
74	793.694.684.675.745.959.547.576.486.496.397	33
73	15.914.477.785.786.564	17
73	253.270.603.194.294.080.492.491.306.072.352	33
72	8.326.337.287.030.818	16
72	69.327.892.617.399.755.799.371.629.872.396	32
71	7.119.655.307.820.645	16
71	50.689.491.702.178.711.787.120.719.498.605	32
70	4.893.930.825.891.954	16
70	23.950.558.928.615.522.551.682.985.505.932	32
69	2.633.793.004.698.078	16
69	6.936.865.591.596.540.456.951.955.686.396	31
68	1.584.019.052.709.082	16
68	2.509.116.359.345.383.835.439.536.119.052	31
67	1.455.445.544.554.452	16
67	2.118.321.733.163.411.143.613.371.238.112	31
66	1.107.790.500.876.913	16
66	1.227.199.793.833.126.213.383.979.917.221	31
65	1.099.608.221.780.343	16
65	1.209.138.241.406.932.396.041.428.319.021	31
64	882.506.981.544.639	15
64	778.818.572.475.033.330.574.275.818.877	30
63	266.763.018.566.987	15
63	71.162.508.074.971.717.947.080.526.117	29
62	242.077.790.404.381	15
62	58.601.656.607.068.386.070.665.610.685	29
61	240.901.135.079.215	15
61	58.033.356.882.455.155.428.865.333.085	29
60	74.920.057.059.261	14
60	5.613.014.949.763.223.679.494.103.165	28
59	71.136.750.663.645	14
59	5.060.437.294.981.881.894.927.340.605	28
58	34.727.947.714.603	14
58	1.206.030.352.468.338.642.530.306.021	28
57	26.372.438.389.158	14
57	695.505.506.590.040.095.605.505.596	27
56	22.546.151.203.551	14
56	508.328.934.093.474.390.439.823.805	27
55	14.554.455.445.544	14
55	211.832.173.316.383.613.371.238.112	27
54	10.960.402.230.853	14
54	120.130.417.062.131.260.714.031.021	27
53	8.471.752.832.059	13
53	71.770.596.047.533.574.069.507.717	26
52	2.237.919.705.745	12
52	5.008.284.609.370.739.064.828.005	25
51	164.555.655.724	12
51	27.078.563.831.413.836.587.072	23
50	145.544.554.452	12
50	21.183.217.331.213.371.238.112	23
49	27.056.705.669	11
49	732.065.321.767.123.560.237	21
48	26.267.929.988	11
48	690.004.145.959.541.400.096	21
47	5.215.527.324	10
47	27.201.725.288.252.710.272	20
46	3.495.950.333	10
46	12.221.668.744.786.612.221	20
45	3.476.828.553	10
45	12.088.336.800.863.388.021	20
44	2.718.194.777	10
44	7.388.582.856.582.858.837	19
43	2.241.769.201	10
43	5.025.529.159.519.255.205	19
42	1.667.309.574	10
42	2.779.921.222.221.299.772	19
41	1.583.998.414	10
41	2.509.050.981.890.509.052	19
40	1.455.445.544	10
40	2.118.321.737.371.238.112	19
39	1.106.902.513	10
39	1.225.233.177.713.325.221	19
38	486.849.854	9
38	237.022.782.287.220.732	18
37	282.442.347	9
37	79.773.680.508.637.797	17
36	277.945.157	9
36	77.253.511.411.535.277	17
35	223.674.495	9
35	50.030.280.608.203.005	17
34	Prime! 110.651.063	9
34	12.243.658.185.634.221	17
33	85.559.327	8
33	7.320.398.778.930.237	16
32	14.554.452	8
32	211.832.131.238.112	15
31	8.308.388	7
31	69.029.344.392.096	14
30	7.395.091	7
30	54.687.400.478.645	14
29	5.040.882	7
29	25.410.511.501.452	14
28	3.504.083	7
28	12.278.611.687.221	14
27	2.705.669	7
27	7.320.655.560.237	13
26	1.659.022	7
26	2.752.360.632.572	13
25	281.349	6
25	79.158.385.197	11
24	266.737	6
24	71.149.694.117	11
23	262.808	6
23	69.069.096.096	11
22	145.544	6
22	21.183.638.112	11
21	73.491	5
21	5.401.221.045	10
20	23.255	5
20	540.888.045	9
19	11.223	5
19	126.000.621	9
18	11.063	5
18	122.434.221	9
17	8.459	4
17	71.588.517	8
16	8.338	4
16	69.555.596	8
15	5.412	4
15	29.311.392	8
14	1.582	4
14	2.509.052	7
13	1.452	4
13	2.114.112	7
12	Prime! 1.123	4
12	1.265.621	7
11	524	3
11	276.672	6
10	Prime! 269	3
10	73.437	5
9	235	3
9	56.165	5
8	144	3
8	21.312	5
7	44	2
7	2.112	4
6	33	2
6	1.221	4
5	21	2
5	525	3
4	14	2
4	252	3
3	Prime! 7	1
3	77	2
2	1	1
2	Prime! 5	1
1	0	1
1	0	1

	[QP5] Case X = 5
One can find the regular numbers of the form n(n+5) at %N n(n+5). under A028557. The palindromic numbers of the form n(n+5) are categorised as follows : %N n(n+5) is a palindrome. under A028558. %N Palindromes of the form n(n+5). under A028559. More terms from Robert Xiao, Nov 11, 2022. See Index Nrs 53 up to 58. More terms from Robert Xiao, Dec 6, 2022. See Index Nrs 59 up to 71. More terms from Robert Xiao, Jul 12, 2023. See Index Nrs 72 & 73.
73	2.506.231.419.648.940.424.963.511.461	28
73	6.281.195.928.835.543.325.626.652.391.932.566.265.233.455.388.295.911.826	55
72	836.015.303.432.690.140.810.586.696	27
72	698.921.587.573.652.967.536.971.366.663.179.635.769.256.375.785.129.896	54
71	25.753.488.991.546.869.297.198.089	26
71	663.242.195.237.725.782.931.112.979.211.139.287.527.732.591.242.366	51
70	25.041.619.383.542.569.343.932.209	26
70	627.082.701.350.214.930.688.563.303.365.886.039.412.053.107.280.726	51
69	21.562.434.345.569.231.068.045.427	26
69	464.938.574.906.983.594.089.051.626.150.980.495.389.609.475.839.464	51
68	6.872.085.043.310.855.129.817.253	25
68	47.225.552.842.496.757.625.410.600.601.452.675.769.424.825.552.274	50
67	6.759.097.559.109.561.670.239.433	25
67	45.685.399.813.560.834.516.762.666.626.761.543.806.531.899.358.654	50
66	2.147.079.595.027.983.211.229.827	25
66	4.609.950.787.385.528.388.669.066.609.668.838.255.837.870.599.064	49
65	780.164.778.439.956.654.091.969	24
65	608.657.081.518.266.654.932.133.331.239.456.662.815.180.756.806	48
64	641.254.613.159.073.918.463.718	24
64	411.207.478.897.793.537.070.997.799.070.735.397.798.874.702.114	48
63	Prime! 260.673.810.645.196.148.291.459	24
63	67.950.835.556.287.576.761.198.589.116.767.578.265.553.805.976	47
62	208.643.994.760.175.228.295.682	24
62	43.532.316.549.484.028.120.813.231.802.182.048.494.561.323.534	47
61	204.249.963.811.643.031.180.677	24
61	41.718.047.717.057.487.834.487.678.443.878.475.071.774.081.714	47
60	63.375.014.555.374.926.087.112	23
60	4.016.392.469.893.983.740.481.001.840.473.893.989.642.936.104	46
59	21.177.084.145.495.595.342.348	23
59	448.468.892.905.400.909.558.898.855.909.004.509.298.864.844	45
58	8.100.597.252.717.522.101.544	22
58	65.619.675.850.734.666.632.588.523.666.643.705.857.691.656	44
57	2.497.635.790.291.580.663.409	22
57	6.238.184.540.945.448.701.302.031.078.445.490.454.818.326	43
56	250.793.918.192.513.995.461	21
56	62.897.589.402.353.402.421.312.420.435.320.498.579.826	41
55	209.241.957.083.066.673.513	21
55	43.782.196.603.951.915.662.626.651.915.930.669.128.734	41
54	83.130.209.419.288.066.596	20
54	6.910.631.718.094.690.390.880.930.964.908.171.360.196	40
53	Prime! 67.372.940.594.612.378.237	20
53	4.539.113.124.365.168.518.338.158.615.634.213.119.354	40
52	21.101.628.452.340.870.048	20
52	445.278.723.340.641.742.616.247.146.043.327.872.544	39
51	20.076.590.060.561.403.787	20
51	403.069.468.459.832.951.080.159.238.954.864.960.304	39
50	7.869.324.084.711.774.564	19
50	61.926.261.550.224.808.533.580.842.205.516.262.916	38
49	6.700.829.702.909.744.672	19
49	44.901.118.707.397.497.077.079.479.370.781.110.944	38
48	2.569.010.347.238.204.644	19
48	6.599.814.164.216.960.812.180.696.124.614.189.956	37
47	2.111.446.108.819.027.072	19
47	4.458.204.670.447.010.722.270.107.440.764.028.544	37
46	251.242.410.444.831.416	18
46	63.122.748.806.129.135.453.192.160.884.722.136	35
45	68.277.287.222.041.227	17
45	4.661.787.950.401.114.554.111.040.597.871.664	34
44	25.870.618.469.371.914	17
44	669.288.899.987.807.323.708.789.998.882.966	33
43	21.854.071.706.057.617	17
43	477.600.450.133.508.191.805.331.054.006.774	33
42	21.654.046.835.524.843	17
42	468.897.744.355.103.575.301.553.447.798.864	33
41	20.441.623.013.274.302	17
41	417.859.951.416.825.656.528.614.159.958.714	33
40	2.638.211.511.329.971	16
40	6.960.159.978.513.982.893.158.799.510.696	31
39	2.602.627.310.579.936	16
39	6.773.668.917.776.563.656.777.198.663.776	31
38	2.169.155.049.742.678	16
38	4.705.233.629.824.170.714.289.263.325.074	31
37	638.147.557.660.608	15
37	407.232.305.348.202.202.843.503.232.704	30
36	634.272.577.528.887	15
36	402.301.702.605.141.141.506.207.103.204	30
35	204.040.586.040.402	15
35	41.632.560.751.711.711.715.706.523.614	29
34	22.104.542.245.332	14
34	488.610.787.875.777.578.787.016.884	27
33	7.989.716.477.391	13
33	63.835.569.389.133.198.396.553.836	26
32	2.546.322.790.449	13
32	6.483.759.753.172.713.579.573.846	25
31	2.471.952.825.156	13
31	6.110.550.769.809.089.670.550.116	25
30	2.228.557.243.297	13
30	4.966.467.386.662.666.837.646.694	25
29	2.048.192.235.518	13
29	4.195.091.433.646.463.341.905.914	25
28	2.008.491.060.208	13
28	4.034.036.338.945.498.336.304.304	25
27	81.294.635.989	11
27	6.608.817.840.990.487.188.066	22
26	26.331.005.924	11
26	693.321.873.101.378.123.396	21
25	22.222.222.222	11
25	493.827.160.595.061.728.394	21
24	8.161.664.181	10
24	66.612.762.244.226.721.666	20
23	249.281.986	9
23	62.141.509.790.514.126	17
22	211.313.148	9
22	44.653.247.574.235.644	17
21	79.585.891	8
21	6.333.914.444.193.336	16
20	6.887.078	7
20	47.431.877.813.474	14
19	2.233.097	7
19	4.986.733.376.894	13
18	2.187.142	7
18	4.783.601.063.874	13
17	2.040.402	7
17	4.163.250.523.614	13
16	207.067	6
16	42.877.777.824	11
15	82.566	5
15	6.817.557.186	10
14	64.493	5
14	4.159.669.514	10
13	25.116	5
13	630.939.036	9
12	20.402	5
12	416.343.614	9
11	Prime! 2.207	4
11	4.881.884	7
10	2.178	4
10	4.754.574	7
9	2.067	4
9	4.282.824	7
8	814	3
8	666.666	6
7	203	3
7	42.224	5
6	202	3
6	41.814	5
5	24	2
5	696	3
4	18	2
4	414	3
3	6	1
3	66	2
2	1	1
2	6	1
1	0	1
1	0	1

	[QP6] Case X = 6
One can find the regular numbers of the form n(n+6) at %N n(n+6). [= m^2 – 9] under A028560. The palindromic numbers of the form n(n+6) are categorised as follows : %N n(n+6) is a palindrome. under A028561. %N Palindromes of the form n(n+6). under A028562. More terms from Robert Xiao, Nov 11, 2022. See Index Nrs 65 up to 80. More terms from Robert Xiao, Dec 6, 2022. See Index Nrs 81 up to 92. More terms from Patrick De Geest, Jul 12, 2023. See Index Nrs 93 and 94.
94	2.795.165.218.974.281.592.995.848.483	28
94	7.812.948.601.363.543.567.508.480.835.380.848.057.653.453.631.068.492.187	55
93	851.393.537.512.371.018.127.715.703	27
93	724.870.955.717.829.116.022.618.045.540.816.220.611.928.717.559.078.427	54
92	88.744.889.877.289.847.258.152.711	26
92	7.875.655.479.332.302.001.140.910.000.190.411.002.032.339.745.565.787	52
91	84.068.905.399.478.083.675.127.201	26
91	7.067.580.855.066.395.291.715.378.888.735.171.925.936.605.580.857.607	52
90	23.955.014.121.817.736.905.377.869	26
90	573.842.701.576.487.200.872.848.222.848.278.002.784.675.107.248.375	51
89	17.195.192.145.745.578.985.612.946	26
89	295.674.632.929.110.448.859.333.585.333.958.844.011.929.236.476.592	51
88	15.254.363.156.489.816.345.136.326	26
88	232.695.595.310.073.953.153.148.313.841.351.359.370.013.595.596.232	51
87	15.197.299.400.626.548.801.068.318	26
87	230.957.909.072.284.059.437.485.191.584.734.950.482.270.909.759.032	51
86	7.826.500.938.840.285.759.047.232	25
86	61.254.116.945.667.874.407.448.533.584.470.447.876.654.961.145.216	50
85	5.468.678.341.743.011.249.449.248	25
85	29.906.442.805.449.091.335.516.999.961.553.319.094.450.824.460.992	50
84	1.661.875.331.028.022.193.991.206	25
84	2.761.829.615.879.498.346.829.758.579.286.438.949.785.169.281.672	49
83	545.708.513.063.786.379.862.346	24
83	297.797.781.230.288.710.012.739.937.210.017.882.032.187.797.792	48
82	479.716.443.882.352.735.470.676	24
82	230.127.866.531.130.481.241.174.471.142.184.031.135.668.721.032	48
81	265.334.149.567.822.968.058.693	24
81	70.402.210.926.879.849.347.439.493.474.394.897.862.901.220.407	47
80	8.489.398.568.713.669.279.491	22
80	72.069.888.058.477.696.543.233.234.569.677.485.088.896.027	44
79	7.851.936.079.637.352.026.472	22
79	61.652.900.198.710.788.984.811.848.988.701.789.100.925.616	44
78	7.717.153.074.043.655.628.595	22
78	59.554.451.568.221.443.812.866.821.834.412.286.515.445.595	44
77	7.575.168.977.459.922.625.369	22
77	57.383.185.037.071.209.736.288.263.790.217.073.058.138.375	44
76	5.451.593.204.571.478.134.648	22
76	29.719.868.468.129.918.246.522.564.281.992.186.486.891.792	44
75	4.426.845.277.698.974.322.357	22
75	19.596.959.112.685.709.084.855.848.090.758.621.195.969.591	44
74	2.732.045.911.701.761.556.713	22
74	7.464.074.863.646.309.504.538.354.059.036.463.684.704.647	43
73	2.314.904.206.645.120.403.385	22
73	5.358.781.485.943.274.306.774.776.034.723.495.841.878.535	43
72	863.127.468.965.137.918.113	21
72	744.989.027.682.165.120.000.000.021.561.286.720.989.447	42
71	509.209.504.685.980.952.716	21
71	259.294.319.662.542.057.845.548.750.245.266.913.492.952	42
70	160.537.294.456.811.922.678	21
70	25.772.222.911.513.136.076.967.063.131.511.922.227.752	41
69	153.769.820.116.504.861.126	21
69	23.645.157.578.662.263.063.436.036.226.687.575.154.632	41
68	145.512.540.528.902.510.236	21
68	21.173.899.451.175.495.634.243.659.457.115.499.837.112	41
67	84.206.785.933.147.216.443	20
67	7.090.782.797.190.879.927.997.299.780.917.972.870.907	40
66	83.876.170.059.195.750.351	20
66	7.035.211.903.799.125.643.443.465.219.973.091.125.307	40
65	Prime! 77.282.200.648.940.778.199	20
65	5.972.538.537.143.142.439.449.342.413.417.358.352.795	40
64	14.078.738.935.233.526.227	20
64	198.210.890.006.460.443.878.344.064.600.098.012.891	39
63	4.471.112.010.115.680.897	19
63	19.990.842.607.000.684.622.648.600.070.624.809.991	38
62	2.658.438.045.166.145.293	19
62	7.067.292.839.986.795.976.795.976.899.382.927.607	37
61	2.311.662.135.070.284.535	19
61	5.343.781.826.717.706.435.346.077.176.281.873.435	37
60	782.618.595.888.768.182	18
60	612.491.866.630.907.042.240.709.036.668.194.216	36
59	782.345.167.260.793.262	18
59	612.063.960.736.318.589.985.813.637.069.360.216	36
58	279.573.353.506.965.483	18
58	78.161.259.991.130.690.809.603.119.995.216.187	35
57	165.225.528.364.325.588	18
57	27.299.475.223.270.560.406.507.232.257.499.272	35
56	24.430.222.115.159.049	17
56	596.835.752.596.006.424.600.695.257.538.695	33
55	15.870.259.146.035.628	17
55	251.865.125.362.327.595.723.263.521.568.152	33
54	13.999.785.814.483.857	17
54	195.994.002.851.423.515.324.158.200.499.591	33
53	7.325.505.333.667.735	16
53	53.663.028.393.594.477.449.539.382.036.635	32
52	2.762.947.364.366.121	16
52	7.633.878.138.257.711.177.528.318.783.367	31
51	2.663.237.144.235.051	16
51	7.092.832.086.433.285.823.346.802.382.907	31
50	1.667.449.597.470.606	16
50	2.780.388.160.104.895.984.010.618.830.872	31
49	1.391.554.098.835.917	16
49	1.936.422.809.987.049.407.899.082.246.391	31
48	716.255.983.645.015	15
48	513.022.634.107.292.292.701.436.220.315	30
47	525.758.362.123.456	15
47	276.421.855.342.742.247.243.558.124.672	30
46	440.309.354.779.717	15
46	193.872.327.906.533.335.609.723.278.391	30
45	279.399.473.798.433	15
45	78.064.065.958.842.924.885.956.046.087	29
44	85.263.388.929.191	14
44	7.269.845.491.691.001.961.945.489.627	28
43	26.528.700.343.143	14
43	703.771.941.896.434.698.149.177.307	27
42	15.977.390.443.078	14
42	255.277.005.370.656.073.500.772.552	27
41	15.849.612.021.116	14
41	251.210.201.219.999.912.102.012.152	27
40	1.656.680.099.188	13
40	2.744.588.951.055.501.598.854.472	25
39	1.389.555.242.017	13
39	1.930.863.770.625.260.773.680.391	25
38	849.022.995.991	12
38	720.840.047.726.627.740.048.027	24
37	436.181.402.107	12
37	190.254.215.546.645.512.452.091	24
36	247.041.361.042	12
36	61.029.434.066.966.043.492.016	23
35	88.440.105.033	11
35	7.821.652.178.778.712.561.287	22
34	24.698.565.242	11
34	610.019.125.161.521.910.016	21
33	7.815.370.552	10
33	61.080.016.911.961.008.016	20
32	2.352.019.439	10
32	5.531.995.455.545.991.355	19
31	1.414.071.397	10
31	1.999.597.924.297.959.991	19
30	782.357.262	9
30	612.082.890.098.280.216	18
29	440.021.417	9
29	193.618.850.058.816.391	18
28	272.515.513	9
28	74.264.706.460.746.247	17
27	235.769.755	9
27	55.587.378.787.378.555	17
26	235.680.755	9
26	55.545.419.691.454.555	17
25	88.837.611	8
25	7.892.121.661.212.987	16
24	84.890.503	8
24	7.206.398.008.936.027	16
23	73.114.035	8
23	5.345.662.552.665.435	16
22	23.975.475	8
22	574.823.545.328.475	15
21	23.887.419	8
21	570.608.929.806.075	15
20	14.529.486	8
20	211.106.050.601.112	15
19	7.560.069	7
19	57.154.688.645.175	14
18	7.153.679	7
18	51.175.166.157.115	14
17	5.401.396	7
17	29.175.111.157.192	14
16	2.436.099	7
16	5.934.592.954.395	13
15	2.273.915	7
15	5.170.703.070.715	13
14	1.731.746	7
14	2.998.954.598.992	13
13	1.378.507	7
13	1.900.289.820.091	13
12	877.173	6
12	769.437.734.967	12
11	486.618	6
11	236.799.997.632	12
10	8.541	4
10	72.999.927	8
9	Prime! 2.731	4
9	7.474.747	7
8	Prime! 863	3
8	749.947	6
7	715	3
7	515.515	6
6	273	3
6	76.167	5
5	Prime! 137	3
5	19.591	5
4	22	2
4	616	3
3	Prime! 5	1
3	55	2
2	1	1
2	Prime! 7	1
1	0	1
1	0	1

	[QP7] Case X = 7
One can find the regular numbers of the form n(n+7) at %N n(n+7). under A028563. The palindromic numbers of the form n(n+7) are categorised as follows : %N n(n+7) is a palindrome. under A028564. %N Palindromes of the form n(n+7). under A028565. More terms from Robert Xiao, Nov 11, 2022. See Index Nrs 98 up to 107. More terms from Robert Xiao, Dec 6, 2022. See Index Nrs 108 up to 137. More terms from Patrick De Geest, Jun 18, 2023. See Index Nrs 138 up to 145. More terms from Patrick De Geest, Jul 12, 2023. See Index Nrs 146 up to 153.
153	2.100.000.000.000.000.000.000.000.009	28
153	4.410.000.000.000.000.000.000.000.052.500.000.000.000.000.000.000.000.144	55
152	2.099.009.901.100.110.010.990.099.004	28
152	4.405.842.564.916.293.609.524.895.273.725.984.259.063.926.194.652.485.044	55
151	2.099.009.901.098.999.010.990.099.004	28
151	4.405.842.564.911.629.609.524.652.063.602.564.259.069.261.194.652.485.044	55
150	2.099.009.901.098.889.010.990.099.004	28
150	4.405.842.564.911.167.827.346.410.295.920.146.437.287.611.194.652.485.044	55
149	2.099.009.890.100.110.009.890.099.004	28
149	4.405.842.518.738.075.901.704.653.273.723.564.071.095.708.378.152.485.044	55
148	904.269.573.462.541.386.266.538.827	27
148	817.703.461.490.126.532.922.086.798.897.680.229.235.621.094.164.307.718	54
147	900.000.000.000.000.000.000.000.002	27
147	810.000.000.000.000.000.000.000.009.900.000.000.000.000.000.000.000.018	54
146	703.555.753.132.391.228.492.062.094	27
146	494.990.697.765.686.229.922.183.044.440.381.229.922.686.567.796.099.494	54
145	290.620.079.487.304.059.362.118.761	27
145	84.460.030.601.206.929.695.144.536.163.544.159.692.960.210.603.006.448	53
144	289.869.916.899.901.911.950.035.161	27
144	84.024.568.723.556.040.060.215.629.792.651.206.004.065.532.786.542.048	53
143	223.248.516.497.058.238.858.537.259	27
143	49.839.900.118.137.284.628.550.848.384.805.582.648.273.181.100.993.894	53
142	223.240.839.793.321.958.143.757.659	27
142	49.836.472.551.627.640.695.630.415.651.403.659.604.672.615.527.463.894	53
141	222.345.984.490.842.989.676.848.574	27
141	49.437.736.819.202.191.299.328.318.481.382.399.219.120.291.863.773.494	53
140	211.361.681.548.300.167.418.659.509	27
140	44.673.760.426.925.051.457.011.684.248.611.075.415.052.962.406.737.644	53
139	210.344.602.961.799.200.853.815.644	27
139	44.244.851.995.156.945.140.085.527.872.558.004.154.965.159.915.844.244	53
138	210.000.000.000.000.000.000.000.009	27
138	44.100.000.000.000.000.000.000.005.250.000.000.000.000.000.000.000.144	53
137	92.844.574.020.302.659.684.404.741	26
137	8.620.114.925.011.459.579.343.263.883.623.439.759.541.105.294.110.268	52
136	90.000.000.000.000.000.000.000.002	26
136	8.100.000.000.000.000.000.000.000.990.000.000.000.000.000.000.000.018	52
135	89.942.697.767.867.255.540.892.201	26
135	8.089.688.881.761.913.392.292.199.449.912.922.933.191.671.888.869.808	52
134	29.106.719.653.300.483.162.096.532	26
134	847.201.128.975.828.598.728.271.919.172.827.895.828.579.821.102.748	51
133	22.196.254.501.377.213.914.954.254	26
133	492.673.713.889.908.231.116.428.828.824.611.132.809.988.317.376.294	51
132	21.143.775.493.851.004.088.309.969	26
132	447.059.242.134.374.271.823.426.616.624.328.172.473.431.242.950.744	51
131	21.000.000.000.000.000.000.000.009	26
131	441.000.000.000.000.000.000.000.525.000.000.000.000.000.000.000.144	51
130	9.000.000.000.000.000.000.000.002	25
130	81.000.000.000.000.000.000.000.099.000.000.000.000.000.000.000.018	50
129	2.959.383.058.147.777.673.802.506	25
129	8.757.948.084.852.092.852.425.432.345.242.582.902.584.808.497.578	49
128	2.112.418.960.652.782.542.134.284	25
128	4.462.313.865.325.382.037.949.060.609.497.302.835.235.683.132.644	49
127	2.100.000.000.000.000.000.000.009	25
127	4.410.000.000.000.000.000.000.052.500.000.000.000.000.000.000.144	49
126	2.099.010.989.988.899.901.099.004	25
126	4.405.847.136.094.181.640.833.582.853.380.461.814.906.317.485.044	49
125	2.098.899.901.099.010.989.989.004	25
125	4.405.380.794.833.438.115.181.482.841.815.118.343.384.970.835.044	49
124	900.000.000.000.000.000.000.002	24
124	810.000.000.000.000.000.000.009.900.000.000.000.000.000.000.018	48
123	663.696.840.102.960.693.723.764	24
123	440.493.495.562.654.974.147.945.549.741.479.456.265.594.394.044	48
122	299.118.880.510.507.612.179.407	24
122	89.472.104.677.859.330.633.777.277.733.603.395.877.640.127.498	47
121	223.056.553.968.123.938.372.679	24
121	49.754.226.268.134.587.077.653.935.677.078.543.186.262.245.794	47
120	210.000.000.000.000.000.000.009	24
120	44.100.000.000.000.000.000.005.250.000.000.000.000.000.000.144	47
119	209.900.990.111.110.990.099.004	24
119	44.058.425.649.624.713.655.669.396.655.631.742.694.652.485.044	47
118	209.900.990.110.010.990.099.004	24
118	44.058.425.649.162.931.477.426.162.477.413.926.194.652.485.044	47
117	209.900.989.010.009.890.099.004	24
117	44.058.425.187.380.292.663.226.162.236.629.208.378.152.485.044	47
116	209.900.988.998.899.890.099.004	24
116	44.058.425.182.716.292.687.547.174.578.629.261.728.152.485.044	47
115	209.889.990.098.900.989.989.004	24
115	44.053.807.943.716.755.609.347.174.390.655.761.734.970.835.044	47
114	90.000.000.000.000.000.000.002	23
114	8.100.000.000.000.000.000.000.990.000.000.000.000.000.000.018	46
113	70.649.133.763.849.917.334.399	23
113	4.991.300.101.582.358.387.061.111.607.838.532.851.010.031.994	46
112	28.848.921.000.940.241.999.747	23
112	832.260.242.918.490.934.344.252.443.439.094.819.242.062.238	45
111	28.320.994.222.093.988.317.817	23
111	802.078.713.727.881.070.495.868.594.070.188.727.317.870.208	45
110	21.000.000.000.000.000.000.009	23
110	441.000.000.000.000.000.000.525.000.000.000.000.000.000.144	45
109	20.990.099.011.110.990.099.004	23
109	440.584.256.496.242.564.455.828.554.465.242.694.652.485.044	45
108	20.990.099.009.900.990.099.004	23
108	440.584.256.445.446.524.850.404.058.425.644.544.652.485.044	45
107	9.000.000.000.000.000.000.002	22
107	81.000.000.000.000.000.000.099.000.000.000.000.000.000.018	44
106	2.961.655.202.402.448.134.306	22
106	8.771.401.537.917.486.025.187.815.206.847.197.351.041.778	43
105	2.100.000.000.000.000.000.009	22
105	4.410.000.000.000.000.000.052.500.000.000.000.000.000.144	43
104	2.099.010.988.998.901.099.004	22
104	4.405.847.131.938.144.910.481.840.194.418.391.317.485.044	43
103	931.598.009.210.422.894.377	21
103	867.874.850.764.823.179.950.059.971.328.467.058.478.768	42
102	900.000.000.000.000.000.002	21
102	810.000.000.000.000.000.009.900.000.000.000.000.000.018	42
101	210.000.000.000.000.000.009	21
101	44.100.000.000.000.000.005.250.000.000.000.000.000.144	41
100	Prime! 209.889.009.900.990.110.989	21
100	44.053.396.477.217.924.841.814.842.971.277.469.335.044	41
99	90.332.071.162.513.227.366	20
99	8.159.883.080.509.353.812.772.183.539.050.803.889.518	40
98	90.000.000.000.000.000.002	20
98	8.100.000.000.000.000.000.990.000.000.000.000.000.018	40
97	28.736.854.321.862.795.056	20
97	825.806.796.315.964.402.909.204.469.513.697.608.528	39
96	22.304.559.111.550.001.839	20
96	497.493.357.160.628.207.535.702.826.061.753.394.794	39
95	22.183.289.860.875.017.854	20
95	492.098.349.051.600.369.131.963.006.150.943.890.294	39
94	21.109.211.394.290.929.329	20
94	445.598.805.688.862.000.797.000.268.886.508.895.544	39
93	21.000.000.000.000.000.009	20
93	441.000.000.000.000.000.525.000.000.000.000.000.144	39
92	20.990.098.888.890.099.004	20
92	440.584.251.365.385.368.919.863.583.563.152.485.044	39
91	9.000.000.000.000.000.002	19
91	81.000.000.000.000.000.099.000.000.000.000.000.018	38
90	2.940.416.984.997.499.916	19
90	8.646.052.045.661.787.666.667.871.665.402.506.468	37
89	2.110.423.522.968.524.904	19
89	4.453.887.446.298.879.977.799.788.926.447.883.544	37
88	2.100.000.000.000.000.009	19
88	4.410.000.000.000.000.052.500.000.000.000.000.144	37
87	2.099.009.888.890.099.004	19
87	4.405.842.513.658.425.780.875.248.563.152.485.044	37
86	2.098.890.099.009.889.004	19
86	4.405.339.647.721.741.680.861.471.277.469.335.044	37
85	900.000.000.000.000.002	18
85	810.000.000.000.000.009.900.000.000.000.000.018	36
84	210.000.000.000.000.009	18
84	44.100.000.000.000.005.250.000.000.000.000.144	35
83	90.000.000.000.000.002	17
83	8.100.000.000.000.000.990.000.000.000.000.018	34
82	66.891.629.092.892.824	17
82	4.474.490.042.701.146.116.411.072.400.944.744	34
81	21.000.000.000.000.009	17
81	441.000.000.000.000.525.000.000.000.000.144	33
80	9.000.000.000.000.002	16
80	81.000.000.000.000.099.000.000.000.000.018	32
79	2.219.536.203.158.254	16
79	4.926.340.957.130.173.710.317.590.436.294	31
78	2.100.000.000.000.009	16
78	4.410.000.000.000.052.500.000.000.000.144	31
77	2.099.198.088.021.029	16
77	4.406.632.612.751.158.511.572.162.366.044	31
76	2.099.009.780.099.004	16
76	4.405.842.056.951.283.821.596.502.485.044	31
75	945.318.054.069.686	15
75	893.626.223.350.104.401.053.322.626.398	30
74	900.000.000.000.002	15
74	810.000.000.000.009.900.000.000.000.018	30
73	895.895.199.033.817	15
73	802.628.207.651.848.848.156.702.826.208	30
72	667.990.192.880.109	15
72	446.210.897.784.009.900.487.798.012.644	30
71	299.692.501.995.807	15
71	89.815.595.752.508.880.525.759.551.898	29
70	295.580.359.679.162	15
70	87.367.749.028.064.846.082.094.776.378	29
69	294.572.920.786.616	15
69	86.773.205.660.760.006.706.650.237.768	29
68	284.366.947.570.417	15
68	80.864.560.870.518.281.507.806.546.808	29
67	222.266.487.782.099	15
67	49.402.391.590.991.519.909.519.320.494	29
66	210.000.000.000.009	15
66	44.100.000.000.005.250.000.000.000.144	29
65	90.000.000.000.002	14
65	8.100.000.000.000.990.000.000.000.018	28
64	22.231.909.422.374	14
64	494.257.796.564.797.465.697.752.494	27
63	22.199.010.990.214	14
63	492.796.088.943.797.349.880.697.294	27
62	21.000.000.000.009	14
62	441.000.000.000.525.000.000.000.144	27
61	9.000.000.000.002	13
61	81.000.000.000.099.000.000.000.018	26
60	6.673.701.200.529	13
60	44.538.287.713.988.931.778.283.544	26
59	2.948.655.661.416	13
59	8.694.570.209.621.269.020.754.968	25
58	2.221.874.942.959	13
58	4.936.728.262.164.612.628.276.394	25
57	2.220.153.281.414	13
57	4.929.080.592.988.892.950.809.294	25
56	2.219.890.109.779	13
56	4.927.912.099.510.159.902.197.294	25
55	2.100.000.000.009	13
55	4.410.000.000.052.500.000.000.144	25
54	900.000.000.002	12
54	810.000.000.009.900.000.000.018	24
53	291.222.254.257	12
53	84.810.401.376.567.310.401.848	23
52	284.038.530.776	12
52	80.677.886.967.376.968.877.608	23
51	210.000.000.009	12
51	44.100.000.005.250.000.000.144	23
50	209.902.199.004	12
50	44.058.933.148.184.133.985.044	23
49	92.917.907.016	11
49	8.633.737.444.884.447.373.368	22
48	90.000.000.002	11
48	8.100.000.000.990.000.000.018	22
47	Prime! 29.097.613.057	11
47	846.671.085.818.580.176.648	21
46	22.199.009.779	11
46	492.796.035.323.530.697.294	21
45	21.000.000.009	11
45	441.000.000.525.000.000.144	21
44	9.000.000.002	10
44	81.000.000.099.000.000.018	20
43	2.984.477.007	10
43	8.907.103.026.203.017.098	19
42	2.219.890.214	10
42	4.927.912.577.752.197.294	19
41	2.100.000.009	10
41	4.410.000.052.500.000.144	19
40	900.000.002	9
40	810.000.009.900.000.018	18
39	703.210.974	9
39	494.505.678.876.505.494	18
38	665.012.044	9
38	442.241.023.320.142.244	18
37	221.990.214	9
37	49.279.656.665.697.294	17
36	210.000.009	9
36	44.100.005.250.000.144	17
35	90.000.002	8
35	8.100.000.990.000.018	16
34	29.857.886	8
34	891.493.565.394.198	15
33	29.802.897	8
33	888.212.878.212.888	15
32	28.867.722	8
32	833.345.575.543.338	15
31	21.155.344	8
31	447.548.727.845.744	15
30	21.152.824	8
30	447.442.111.244.744	15
29	21.117.304	8
29	445.940.676.049.544	15
28	21.037.924	8
28	442.594.393.495.244	15
27	21.000.009	8
27	441.000.525.000.144	15
26	9.000.002	7
26	81.000.099.000.018	14
25	8.972.876	7
25	80.512.566.521.508	14
24	2.219.779	7
24	4.927.434.347.294	13
23	2.109.664	7
23	4.450.696.960.544	13
22	2.100.009	7
22	4.410.052.500.144	13
21	942.997	6
21	889.249.942.988	12
20	900.002	6
20	810.009.900.018	12
19	669.024	6
19	447.597.795.744	12
18	292.967	6
18	85.831.713.858	11
17	223.114	6
17	49.781.418.794	11
16	210.009	6
16	44.105.250.144	11
15	90.002	5
15	8.100.990.018	10
14	29.696	5
14	882.060.288	9
13	21.009	5
13	441.525.144	9
12	9.002	4
12	81.099.018	8
11	2.982	4
11	8.913.198	7
10	2.972	4
10	8.853.588	7
9	2.109	4
9	4.462.644	7
8	902	3
8	819.918	6
7	664	3
7	445.544	6
6	219	3
6	49.494	5
5	26	2
5	858	3
4	Prime! 19	2
4	494	3
3	4	1
3	44	2
2	1	1
2	8	1
1	0	1
1	0	1

	[QP8] Case X = 8
One can find the regular numbers of the form n(n+8) at %N n(n+8). under A028566. The palindromic numbers of the form n(n+8) are categorised as follows : %N n(n+8) is a palindrome. under A028567. %N Palindromes of the form n(n+8). under A028568. More terms from Robert Xiao, Nov 11, 2022. See Index Nrs 109 up to 124. More terms from Robert Xiao, Dec 6, 2022. See Index Nrs 125 up to 163. More terms from Patrick De Geest, Jun 18, 2023. See Index Nrs 164 up to 173. More terms from Patrick De Geest, Jul 12, 2023. See Index Nrs 174 up to 182.
182	Prime! 1.989.001.099.989.001.099.989.001.099	28
182	3.956.125.375.757.456.351.558.787.339.337.878.551.536.547.575.735.216.593	55
181	1.989.001.099.891.089.109.989.001.099	28
181	3.956.125.375.367.962.239.944.149.945.499.414.499.322.697.635.735.216.593	55
180	1.989.000.011.989.889.902.099.998.893	28
180	3.956.121.047.695.782.174.311.255.476.745.521.134.712.875.967.401.216.593	55
179	1.989.000.011.009.780.101.099.998.893	28
179	3.956.121.043.796.905.363.391.053.486.843.501.933.635.096.973.401.216.593	55
178	1.989.000.001.198.889.020.999.998.893	28
178	3.956.121.004.769.180.526.975.330.496.940.335.796.250.819.674.001.216.593	55
177	1.989.000.001.090.090.089.000.001.099	28
177	3.956.121.004.336.378.375.230.300.789.870.030.325.738.736.334.001.216.593	55
176	940.381.269.960.510.574.847.387.748	27
176	884.316.932.892.542.668.448.392.515.515.293.844.866.245.298.239.613.488	54
175	699.999.999.999.999.999.999.999.996	27
175	489.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.984	54
174	696.530.261.845.120.803.003.147.456	27
174	485.154.405.666.032.548.698.852.939.939.258.896.845.230.666.504.451.584	54
173	300.375.685.043.909.859.075.967.661	27
173	90.225.552.165.598.132.998.322.584.548.522.389.923.189.556.125.552.209	53
172	198.900.109.998.901.099.989.001.099	27
172	39.561.253.757.574.957.333.867.843.934.876.833.375.947.575.735.216.593	53
171	198.900.109.989.099.109.989.001.099	27
171	39.561.253.753.675.723.555.547.665.756.674.555.532.757.635.735.216.593	53
170	198.900.109.989.000.109.989.001.099	27
170	39.561.253.753.636.341.333.769.833.933.896.733.314.363.635.735.216.593	53
169	198.900.010.998.901.099.890.001.099	27
169	39.561.214.375.362.978.512.067.843.934.876.021.587.926.357.341.216.593	53
168	198.900.000.001.197.900.000.001.099	27
168	39.561.210.000.476.524.620.001.873.737.810.002.642.567.400.001.216.593	53
167	194.440.811.718.171.043.721.760.329	27
167	37.807.229.261.621.241.871.720.723.032.702.717.814.212.616.292.270.873	53
166	194.439.940.250.862.694.718.969.079	27
166	37.806.890.364.759.054.681.721.422.322.412.718.645.095.746.309.860.873	53
165	188.423.336.257.520.089.865.660.409	27
165	35.503.353.646.414.484.905.331.650.405.613.350.948.441.464.635.330.553	53
164	177.683.642.357.385.378.167.235.473	27
164	31.571.476.761.387.235.308.327.677.777.672.380.353.278.316.767.417.513	53
163	92.050.912.589.669.501.928.089.454	26
163	8.473.370.508.590.975.209.842.907.557.092.489.025.790.958.050.733.748	52
162	69.999.999.999.999.999.999.999.996	26
162	4.899.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.984	52
161	69.695.315.501.021.370.742.714.256	26
161	4.857.437.002.786.909.762.312.922.552.292.132.679.096.872.007.347.584	52
160	58.024.097.611.233.580.084.688.089	26
160	3.366.795.903.597.962.454.588.909.669.098.854.542.697.953.095.976.633	52
159	29.401.379.900.677.093.898.661.474	26
159	864.441.140.063.938.999.885.494.888.494.588.999.839.360.041.144.468	51
158	24.123.530.333.835.499.490.386.645	26
158	581.944.715.767.481.485.488.794.797.497.884.584.184.767.517.449.185	51
157	22.891.225.047.141.858.516.493.767	26
157	524.008.184.158.894.782.660.805.070.508.066.287.498.851.481.800.425	51
156	19.890.991.099.989.000.109.008.893	26
156	395.651.526.939.841.612.532.390.121.093.235.216.148.939.625.156.593	51
155	19.890.010.999.891.099.989.001.099	26
155	395.612.537.575.788.955.166.675.494.576.661.559.887.575.735.216.593	51
154	19.890.010.990.089.099.010.998.893	26
154	395.612.537.185.865.140.715.940.323.049.517.041.568.581.735.216.593	51
153	19.890.000.119.888.902.099.998.893	26
153	395.612.104.769.180.539.911.304.969.403.119.935.081.967.401.216.593	51
152	6.999.999.999.999.999.999.999.996	25
152	48.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.984	49
151	2.909.131.455.622.990.229.437.888	25
151	8.463.045.826.095.137.970.618.993.998.160.797.315.906.285.403.648	49
150	2.385.228.494.909.240.344.660.137	25
150	5.689.314.972.926.999.992.782.216.122.872.999.996.292.794.139.865	49
149	2.339.186.690.327.933.141.865.315	25
149	5.471.794.372.207.349.781.430.012.100.341.879.437.022.734.971.745	49
148	2.247.524.629.901.157.080.866.285	25
148	5.051.366.962.012.333.109.501.542.451.059.013.332.102.696.631.505	49
147	2.213.463.837.872.737.975.571.496	25
147	4.899.422.161.570.310.465.969.880.889.695.640.130.751.612.249.984	49
146	1.989.010.998.909.910.998.901.099	25
146	3.956.164.753.784.601.972.858.857.588.582.791.064.873.574.616.593	49
145	1.989.001.098.909.910.989.001.099	25
145	3.956.125.371.464.833.517.238.857.588.327.153.384.641.735.216.593	49
144	1.989.001.090.090.090.089.001.099	25
144	3.956.125.336.379.566.670.450.897.980.540.766.659.736.335.216.593	49
143	1.989.000.000.119.790.000.001.099	25
143	3.956.121.000.476.524.620.018.737.378.100.264.256.740.001.216.593	49
142	948.792.619.912.543.084.830.041	24
142	900.207.435.600.507.448.645.210.012.546.844.705.006.534.702.009	48
141	909.818.522.781.359.808.085.084	24
141	827.769.744.396.055.736.094.879.978.490.637.550.693.447.967.728	48
140	699.999.999.999.999.999.999.996	24
140	489.999.999.999.999.999.999.999.999.999.999.999.999.999.999.984	48
139	294.173.581.032.401.080.096.324	24
139	86.538.095.777.426.644.521.459.095.412.544.662.477.759.083.568	47
138	229.094.925.856.695.125.655.975	24
138	52.484.485.053.284.636.853.970.707.935.863.648.235.058.448.425	47
137	225.774.460.477.394.629.598.835	24
137	50.974.107.003.858.629.637.061.416.073.692.685.830.070.147.905	47
136	198.909.910.998.900.109.008.893	24
136	39.565.152.693.590.362.561.701.210.716.526.309.539.625.156.593	47
135	198.900.109.008.909.910.998.893	24
135	39.561.253.363.756.245.537.801.210.873.554.265.736.335.216.593	47
134	71.367.064.035.131.893.809.157	23
134	5.093.257.828.994.616.229.073.443.709.226.164.998.287.523.905	46
133	69.999.999.999.999.999.999.996	23
133	4.899.999.999.999.999.999.999.999.999.999.999.999.999.999.984	46
132	69.995.488.827.581.978.530.796	23
132	4.899.368.456.212.153.579.429.669.249.753.512.126.548.639.984	46
131	19.891.099.989.010.999.891.099	23
131	395.655.858.772.833.399.988.595.889.993.338.277.858.556.593	45
130	19.890.563.483.056.915.847.643	23
130	395.634.515.673.517.267.850.626.058.762.715.376.515.436.593	45
129	19.890.100.089.010.899.901.099	23
129	395.616.081.550.871.408.168.797.861.804.178.055.180.616.593	45
128	19.890.000.011.979.000.001.099	23
128	395.612.100.476.524.620.187.373.781.026.425.674.001.216.593	45
127	18.799.220.284.246.541.699.533	23
127	353.410.683.295.626.624.093.636.390.426.626.592.386.014.353	45
126	18.336.810.047.561.706.715.903	23
126	336.238.602.720.359.960.912.737.219.069.953.027.206.832.633	45
125	17.819.832.852.327.570.473.773	23
125	317.546.442.884.892.956.084.020.480.659.298.488.244.645.713	45
124	6.999.999.999.999.999.999.996	22
124	48.999.999.999.999.999.999.999.999.999.999.999.999.999.984	44
123	5.929.875.601.515.954.129.759	22
123	35.163.424.649.454.198.811.899.811.889.145.494.642.436.153	44
122	2.383.913.278.838.667.355.887	22
122	5.683.042.521.023.325.775.770.775.775.233.201.252.403.865	43
121	1.989.109.998.910.999.891.099	22
121	3.956.558.587.767.717.987.969.697.897.177.677.858.556.593	43
120	1.989.010.990.090.098.901.099	22
120	3.956.164.718.699.195.508.969.698.055.919.968.174.616.593	43
119	1.989.001.090.090.089.001.099	22
119	3.956.125.336.379.562.342.789.872.432.659.736.335.216.593	43
118	1.926.073.753.086.323.748.213	22
118	3.709.760.102.328.036.820.410.140.286.308.232.010.679.073	43
117	952.131.696.042.777.579.321	21
117	906.554.766.609.296.194.308.803.491.692.906.667.455.609	42
116	749.798.101.226.134.068.405	21
116	562.197.192.602.315.991.180.081.199.513.206.291.791.265	42
115	699.999.999.999.999.999.996	21
115	489.999.999.999.999.999.999.999.999.999.999.999.999.984	42
114	220.269.975.269.987.113.956	21
114	48.518.862.005.440.734.757.475.743.704.450.026.881.584	41
113	198.909.910.999.890.991.099	21
113	39.565.152.693.984.555.099.999.055.548.939.625.156.593	41
112	198.900.001.197.900.001.099	21
112	39.561.210.476.524.621.873.737.812.642.567.401.216.593	41
111	92.818.011.046.340.572.424	20
111	8.615.183.174.598.600.524.884.250.068.954.713.815.168	40
110	69.999.999.999.999.999.996	20
110	4.899.999.999.999.999.999.999.999.999.999.999.999.984	40
109	56.156.702.135.175.213.223	20
109	3.153.575.194.698.792.352.222.532.978.964.915.753.513	40
108	23.683.958.404.363.815.505	20
108	560.929.885.699.635.409.979.904.536.996.588.929.065	39
107	22.133.746.889.973.981.096	20
107	489.902.751.389.432.880.606.088.234.983.157.209.984	39
106	19.890.991.099.890.991.099	20
106	395.651.526.935.942.619.999.916.249.539.625.156.593	39
105	6.999.999.999.999.999.996	19
105	48.999.999.999.999.999.999.999.999.999.999.999.984	38
104	2.412.063.028.201.142.847	19
104	5.818.048.052.014.867.252.527.684.102.508.408.185	37
103	1.989.000.119.790.001.099	19
103	3.956.121.476.524.638.737.378.364.256.741.216.593	37
102	699.999.999.999.999.996	18
102	489.999.999.999.999.999.999.999.999.999.999.984	36
101	294.011.283.367.553.974	18
101	86.442.634.747.436.122.422.163.474.743.624.468	35
100	238.427.502.869.428.605	18
100	56.847.674.124.551.387.578.315.542.147.674.865	35
99	Prime! 183.952.041.061.237.039	18
99	33.838.353.410.595.039.093.059.501.435.383.833	35
98	74.972.390.516.353.905	17
98	5.620.859.339.736.673.553.766.379.339.580.265	34
97	69.999.999.999.999.996	17
97	4.899.999.999.999.999.999.999.999.999.999.984	34
96	29.440.591.428.026.318	17
96	866.748.423.631.976.949.679.136.324.847.668	33
95	24.197.921.094.983.445	17
95	585.539.385.319.044.999.440.913.583.935.585	33
94	23.672.512.354.237.737	17
94	560.387.841.161.538.474.835.161.148.783.065	33
93	9.108.243.842.099.284	16
93	82.960.105.887.139.599.593.178.850.106.928	32
92	6.999.999.999.999.996	16
92	48.999.999.999.999.999.999.999.999.999.984	32
91	3.004.385.430.296.781	16
91	9.026.331.813.779.597.959.773.181.336.209	31
90	2.913.451.145.839.304	16
90	8.488.197.579.192.376.732.919.757.918.848	31
89	1.835.437.923.830.089	16
89	3.368.832.372.233.722.273.322.732.388.633	31
88	951.856.579.421.421	15
88	906.030.947.787.855.558.787.749.030.609	30
87	930.589.355.492.668	15
87	865.996.548.556.266.662.655.845.699.568	30
86	699.999.999.999.996	15
86	489.999.999.999.999.999.999.999.999.984	30
85	697.090.484.234.136	15
85	485.935.143.209.787.787.902.341.539.584	30
84	630.600.416.802.799	15
84	397.656.885.671.868.868.176.588.656.793	30
83	630.477.098.800.549	15
83	397.501.372.111.962.269.111.273.105.793	30
82	625.328.055.612.043	15
82	391.035.177.135.543.345.531.771.530.193	30
81	294.411.411.576.174	15
81	86.678.079.266.277.677.266.297.087.668	29
80	225.762.237.924.335	15
80	50.968.588.072.605.850.627.088.586.905	29
79	220.050.252.330.346	15
79	48.422.113.550.650.705.605.531.122.484	29
78	183.014.335.582.889	15
78	33.494.247.028.847.774.882.074.249.433	29
77	76.761.240.619.745	14
77	5.892.288.061.483.003.841.608.822.985	28
76	71.108.945.208.785	14
76	5.056.482.088.706.556.078.802.846.505	28
75	69.999.999.999.996	14
75	4.899.999.999.999.999.999.999.999.984	28
74	69.696.500.774.736	14
74	4.857.602.220.243.333.420.222.067.584	28
73	63.220.385.270.749	14
73	3.996.817.113.782.442.873.117.186.993	28
72	58.198.234.728.953	14
72	3.387.034.525.566.776.655.254.307.833	28
71	55.827.247.193.619	14
71	3.116.681.529.217.887.129.251.866.113	28
70	22.388.140.098.985	14
70	501.228.817.091.959.190.718.822.105	27
69	18.956.175.005.059	14
69	359.336.570.822.575.228.075.633.953	27
68	6.999.999.999.996	13
68	48.999.999.999.999.999.999.999.984	26
67	6.933.240.247.186	13
67	48.069.820.325.255.252.302.896.084	26
66	3.010.857.767.521	13
66	9.065.264.496.265.626.944.625.609	25
65	1.830.169.111.639	13
65	3.349.518.977.212.127.798.159.433	25
64	906.998.967.658	12
64	822.647.127.339.933.721.746.228	24
63	722.945.524.917	12
63	522.650.232.003.300.232.056.225	24
62	714.048.799.585	12
62	509.865.688.194.491.886.568.905	24
61	699.999.999.996	12
61	489.999.999.999.999.999.999.984	24
60	228.640.304.575	12
60	52.276.388.877.977.888.367.225	23
59	197.920.018.793	12
59	39.172.333.840.604.833.327.193	23
58	93.910.205.194	11
58	8.819.126.640.330.466.219.188	22
57	93.236.732.768	11
57	8.693.088.337.997.338.803.968	22
56	69.999.999.996	11
56	4.899.999.999.999.999.999.984	22
55	62.884.810.893	11
55	3.954.499.441.551.449.944.593	22
54	Prime! 18.963.885.559	11
54	359.628.955.646.559.826.953	21
53	7.228.936.825	10
53	52.257.527.677.672.575.225	20
52	6.999.999.996	10
52	48.999.999.999.999.999.984	20
51	1.771.260.819	10
51	3.137.364.903.094.637.313	19
50	699.999.996	9
50	489.999.999.999.999.984	18
49	695.334.376	9
49	483.489.900.009.984.384	18
48	287.876.058	9
48	82.872.627.072.627.828	17
47	225.207.185	9
47	50.718.277.977.281.705	17
46	182.901.103	9
46	33.452.814.941.825.433	17
45	91.842.938	8
45	8.435.125.995.215.348	16
44	91.050.784	8
44	8.290.245.995.420.928	16
43	69.999.996	8
43	4.899.999.999.999.984	16
42	69.692.736	8
42	4.857.078.008.707.584	16
41	59.318.009	8
41	3.518.626.666.268.153	16
40	57.524.753	8
40	3.309.097.667.909.033	16
39	23.352.327	8
39	545.331.363.133.545	15
38	Prime! 19.782.199	8
38	391.335.555.533.193	15
37	Prime! 17.869.169	8
37	319.307.343.703.913	15
36	17.646.619	8
36	311.403.303.304.113	15
35	7.243.225	7
35	52.464.366.346.425	14
34	6.999.996	7
34	48.999.999.999.984	14
33	2.281.817	7
33	5.206.707.076.025	13
32	Prime Curios! 1.934.063	7
32	3.740.615.160.473	13
31	896.478	6
31	803.679.976.308	12
30	699.996	6
30	489.999.999.984	12
29	597.883	6
29	357.468.864.753	12
28	298.144	6
28	88.892.229.888	11
27	283.778	6
27	80.532.223.508	11
26	241.097	6
26	58.129.692.185	11
25	228.175	6
25	52.065.656.025	11
24	188.583	6
24	35.565.056.553	11
23	93.844	5
23	8.807.447.088	10
22	69.996	5
22	4.899.999.984	10
21	57.489	5
21	3.305.445.033	10
20	30.031	5
20	902.101.209	9
19	28.314	5
19	801.909.108	9
18	9.394	4
18	88.322.388	8
17	6.996	4
17	48.999.984	8
16	2.828	4
16	8.020.208	7
15	2.235	4
15	5.013.105	7
14	924	3
14	861.168	6
13	707	3
13	505.505	6
12	696	3
12	489.984	6
11	294	3
11	88.788	5
10	284	3
10	82.928	5
9	225	3
9	52.425	5
8	216	3
8	48.384	5
7	Prime! 173	3
7	31.313	5
6	91	2
6	9.009	4
5	88	2
5	8.448	4
4	66	2
4	4.884	4
3	Prime! 3	1
3	33	2
2	1	1
2	9	1
1	0	1
1	0	1

	[QP9] Case X = 9
One can find the regular numbers of the form n(n+9) at %N n(n+9). under A028569. The palindromic numbers of the form n(n+9) are categorised as follows : %N n(n+9) is a palindrome. under A028570. %N Palindromes of the form n(n+9). under A028571. More terms from Robert Xiao, Nov 11, 2022. See Index Nrs 77 up to 85. More terms from Robert Xiao, Dec 6, 2022. See Index Nrs 86 up to 101. More terms from Patrick De Geest, Jun 18, 2023. See Index Nr 102.
102	154.973.205.123.285.989.320.662.649	27
102	24.016.694.306.184.074.808.437.163.436.173.480.847.048.160.349.661.042	53
101	82.707.366.354.715.857.329.288.098	26
101	6.840.508.449.333.184.602.097.766.666.677.902.064.813.339.448.050.486	52
100	52.820.565.065.697.608.033.299.964	26
100	2.790.012.093.859.594.555.251.062.002.601.525.554.959.583.902.100.972	52
99	50.292.512.429.083.793.636.283.012	26
99	2.529.336.806.429.547.865.029.476.446.749.205.687.459.246.086.339.252	52
98	26.216.007.697.705.266.896.416.638	26
98	687.279.059.606.141.808.579.293.404.392.975.808.141.606.950.972.786	51
97	Prime! 15.455.345.455.455.455.454.355.447	26
97	238.867.703.147.467.599.798.062.414.260.897.995.764.741.307.768.832	51
96	8.270.467.307.059.204.094.201.893	25
96	68.400.629.477.135.123.300.078.188.187.000.332.153.177.492.600.486	50
95	1.688.625.387.226.424.383.432.794	25
95	2.851.455.698.385.591.693.252.159.512.523.961.955.838.965.541.582	49
94	1.474.521.334.993.148.692.774.159	25
94	2.174.213.167.349.977.427.645.775.775.467.247.799.437.613.124.712	49
93	488.736.514.976.372.064.428.919	24
93	238.863.381.071.249.555.225.253.352.522.555.942.170.183.368.832	48
92	462.998.101.331.663.807.322.259	24
92	214.367.241.836.725.627.031.276.672.130.726.527.638.142.763.412	48
91	162.822.177.349.330.856.861.334	24
91	26.511.061.436.776.950.336.909.090.963.305.967.763.416.011.562	47
90	146.866.791.500.238.956.506.807	24
90	21.569.854.445.574.661.800.926.962.900.816.647.554.445.896.512	47
89	52.659.272.316.002.441.761.364	23
89	2.772.998.960.850.901.166.609.669.066.611.090.580.698.992.772	46
88	51.791.705.794.512.977.511.757	23
88	2.682.380.789.105.389.131.172.442.711.319.835.019.870.832.862	46
87	50.588.282.870.433.199.266.487	23
87	2.559.174.363.778.965.050.964.884.690.505.698.773.634.719.552	46
86	14.816.822.946.221.160.594.607	23
86	219.538.242.219.665.913.662.030.266.319.566.912.242.835.912	45
85	5.166.268.943.655.378.126.634	22
85	26.690.334.798.178.056.572.577.527.565.087.189.743.309.662	44
84	1.658.664.988.687.590.206.852	22
84	2.751.169.544.698.003.748.881.888.473.008.964.459.611.572	43
83	1.491.718.489.188.799.379.589	22
83	2.225.224.050.987.714.171.538.351.714.177.890.504.225.222	43
82	1.423.357.185.333.067.894.537	22
82	2.025.945.677.039.273.386.687.866.833.729.307.765.495.202	43
81	496.563.057.979.871.313.574	21
81	246.574.870.550.321.039.834.438.930.123.055.078.475.642	42
80	479.220.413.524.301.648.177	21
80	229.652.204.738.402.674.031.130.476.204.837.402.256.922	42
79	251.118.491.446.809.237.028	21
79	63.060.496.746.521.203.936.563.930.212.564.769.406.036	41
78	145.649.860.571.712.833.659	21
78	21.213.881.884.559.388.693.439.688.395.548.818.831.212	41
77	49.662.171.718.171.586.574	20
77	2.466.331.299.765.161.795.775.971.615.679.921.336.642	40
76	25.210.573.274.064.841.328	20
76	635.573.004.806.992.453.404.354.299.608.400.375.536	39
75	16.681.463.061.790.801.327	20
75	278.271.209.881.890.936.121.639.098.188.902.172.872	39
74	15.620.041.971.084.230.442	20
74	243.985.711.178.432.931.060.139.234.871.117.589.342	39
73	15.445.446.545.435.545.544	20
73	238.561.818.987.906.827.999.728.609.789.818.165.832	39
72	15.444.456.545.434.555.544	20
72	238.531.237.983.816.285.595.582.618.389.732.135.832	39
71	15.219.869.175.229.085.444	20
71	231.644.417.711.088.481.737.184.880.117.714.446.132	39
70	1.696.782.851.936.705.269	19
70	2.879.072.046.626.459.090.909.546.266.402.709.782	37
69	1.684.230.962.345.308.797	19
69	2.836.633.934.522.604.993.994.062.254.393.366.382	37
68	1.668.295.293.997.506.039	19
68	2.783.209.187.974.225.124.215.224.797.819.023.872	37
67	1.599.655.235.294.309.487	19
67	2.558.896.871.804.492.662.662.944.081.786.988.552	37
66	1.438.728.178.155.345.529	19
66	2.069.938.770.618.199.676.769.918.160.778.399.602	37
65	262.139.313.895.938.913	18
65	68.717.019.889.833.594.449.533.898.891.071.786	35
64	251.096.226.342.076.228	18
64	63.049.314.883.231.178.087.113.238.841.394.036	35
63	172.610.504.583.658.592	18
63	29.794.386.292.625.225.352.252.629.268.349.792	35
62	50.853.897.306.350.404	17
62	2.586.118.871.244.833.333.384.421.788.116.852	34
61	48.313.653.281.364.572	17
61	2.334.209.093.391.910.110.191.933.909.024.332	34
60	45.108.988.816.777.754	17
60	2.034.820.872.072.180.880.812.702.780.284.302	34
59	25.240.619.955.431.648	17
59	637.088.895.734.534.555.435.437.598.880.736	33
58	16.517.616.586.567.127	17
58	272.831.657.700.837.616.738.007.756.138.272	33
57	1.545.039.109.604.072	16
57	2.387.145.850.206.157.516.020.585.417.832	31
56	1.510.415.941.428.602	16
56	2.281.356.316.121.663.661.216.136.531.822	31
55	458.433.179.591.059	15
55	210.160.980.149.972.279.941.089.061.012	30
54	165.822.864.864.839	15
54	27.497.222.511.984.148.911.522.279.472	29
53	153.465.444.564.347	15
53	23.551.642.675.334.043.357.624.615.532	29
52	147.647.028.000.832	15
52	21.799.644.877.479.797.477.844.699.712	29
51	52.409.339.938.527	14
51	2.746.738.912.792.552.972.198.376.472	28
50	50.840.261.816.212	14
50	2.584.732.221.541.441.451.222.374.852	28
49	16.828.270.461.922	14
49	283.190.686.739.747.937.686.091.382	27
48	4.855.892.951.019	13
48	23.579.696.351.799.715.369.697.532	26
47	4.515.673.445.362	13
47	20.391.306.665.188.156.660.319.302	26
46	2.514.764.874.268	13
46	6.324.042.372.874.782.732.404.236	25
45	544.332.666.724	12
45	296.298.052.067.760.250.892.692	24
44	262.573.714.118	12
44	68.944.955.348.084.355.944.986	23
43	152.730.610.572	12
43	23.326.639.407.070.493.662.332	23
42	47.972.497.144	11
42	2.301.360.482.662.840.631.032	22
41	16.700.003.327	11
41	278.890.111.272.111.098.872	21
40	15.114.678.514	11
40	228.453.506.717.605.354.822	21
39	1.665.597.364	10
39	2.774.214.593.954.124.772	19
38	1.632.565.357	10
38	2.665.269.659.569.625.662	19
37	Prime! 1.534.664.347	10
37	2.355.194.671.764.915.532	19
36	1.458.904.659	10
36	2.128.402.817.182.048.212	19
35	1.417.320.729	10
35	2.008.798.061.608.978.002	19
34	514.668.957	9
34	264.884.139.931.488.462	18
33	497.665.874	9
33	247.671.326.623.176.742	18
32	164.399.727	9
32	27.027.271.717.272.072	17
31	151.698.472	9
31	23.012.427.772.421.032	17
30	54.072.524	8
30	2.923.838.338.383.292	16
29	52.030.352	8
29	2.707.157.997.517.072	16
28	49.407.017	8
28	2.441.053.773.501.442	16
27	48.675.319	8
27	2.369.287.117.829.632	16
26	26.216.753	8
26	687.318.373.813.786	15
25	26.211.438	8
25	687.039.717.930.786	15
24	26.132.578	8
24	682.911.868.119.286	15
23	25.234.083	8
23	636.759.171.957.636	15
22	15.336.644	8
22	235.212.787.212.532	15
21	15.019.614	8
21	225.588.939.885.522	15
20	14.790.532	8
20	218.759.969.957.812	15
19	14.674.184	8
19	215.331.808.133.512	15
18	1.547.147	7
18	2.393.677.763.932	13
17	487.619	6
17	237.776.677.732	12
16	160.187	6
16	25.661.316.652	11
15	16.997	5
15	289.050.982	9
14	15.904	5
14	253.080.352	9
13	5.392	4
13	29.122.192	8
12	1.694	4
12	2.884.882	7
11	1.664	4
11	2.783.872	7
10	1.639	4
10	2.701.072	7
9	258	3
9	68.886	5
8	248	3
8	63.736	5
7	Prime! 167	3
7	29.392	5
6	Prime! 157	3
6	26.062	5
5	Prime! 137	3
5	20.002	5
4	44	2
4	2.332	4
3	12	2
3	252	3
2	Prime! 2	1
2	22	2
1	0	1
1	0	1

Further Topics Revealed

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online :

Neil Sloane's Integer Sequences

That case X=2 with formula n(n+2) equalling n²–1 was already known to me.
I was quite surprised to find an entry for case X = 4.

%N n^2 – 4 denominators of wavelengths of hydrogen lines. under A028347.

Numbers of the form n²–Y.
A while ago I submitted numbers of the form n(n+X) to Sloane's table.
I found that when X equals an even number there is a corresponding formula n²–Y.
whereby Y = (X/2)² or the square of the natural numbers.

n(n+0) = n²–0 (trivial case)

n(n+2) = n²–1

n(n+4) = n²–4

n(n+6) = n²–9

n(n+8) = n²–16

n(n+10) = n²–25

The variables at the left are the even numbers (0,2,4,6,8,10,...).
The variables at the right are the squares of the natural numbers (0,1,4,9,16,25,...).
Obviously all these numbers are composite.
The intermediate Y variables can deliver primes. Here's a summary :

%N n^2 – 2. under A008865

%N n^2 – 2 is prime. under A028870

%N Primes of the form n^2 – 2. under A028871

%N n^2 – 3. under A028872

%N n^2 – 3 is prime. under A028873

%N Primes of the form n^2 – 3. under A028874

%N n^2 – 5. under A028875

%N n^2 – 5 is prime. under A028876

%N Primes of the form n^2 – 5. under A028877

%N n^2 – 6. under A028878

%N n^2 – 6 is prime. under A028879

%N Primes of the form n^2 – 6. under A028880

%N n^2 – 7. under A028881

%N n^2 – 7 is prime. under A028882

%N Primes of the form n^2 – 7. under A028883

%N n^2 – 8. under A028884

%N n^2 – 8 is prime. under A028885

%N Primes of the form n^2 – 8. under A028886

This last sequence is a subset of :

%N Divisible only by primes congruent to 1 mod 8. under A004625

Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.

Contributions

[ November 11, 2022 ]
Robert Xiao (email) added

for Case X = 3 [n(n + 3] entries [72] up to [77].

for Case X = 4 [n(n + 4] entries [86] up to [97].

for Case X = 5 [n(n + 5] entries [53] up to [58].

for Case X = 6 [n(n + 6] entries [65] up to [80].

for Case X = 7 [n(n + 7] entries [98] up to [107].

for Case X = 8 [n(n + 8] entries [109] up to [124].

for Case X = 9 [n(n + 9] entries [77] up to [85].

[ December 6, 2022 ]
Robert Xiao (email) added

for Case X = 3 [n(n + 3] entries [78] up to [92].

for Case X = 4 [n(n + 4] entries [98] up to [114].

for Case X = 5 [n(n + 5] entries [59] up to [71].

for Case X = 6 [n(n + 6] entries [81] up to [92].

for Case X = 7 [n(n + 7] entries [108] up to [137].

for Case X = 8 [n(n + 8] entries [125] up to [163].

for Case X = 9 [n(n + 9] entries [86] up to [101].

[ June 18, 2022 ]
Patrick De Geest (email) added

for Case X = 3 [n(n + 3] entry [93].

for Case X = 4 [n(n + 4] entries [115] up to [119].

for Case X = 7 [n(n + 7] entries [138] up to [145].

for Case X = 8 [n(n + 8] entries [164] up to [173].

for Case X = 9 [n(n + 9] entry [102].

[ July 12, 2022 ]
Patrick De Geest (email) added

for Case X = 3 [n(n + 3] entry [94] up to [96].

for Case X = 4 [n(n + 4] entries [120] up to [124].

for Case X = 5 [n(n + 5] entries [72] and [73].

for Case X = 6 [n(n + 6] entries [93] and [94].

for Case X = 7 [n(n + 7] entries [146] up to [153].

for Case X = 8 [n(n + 8] entries [174] up to [182].

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Patrick De Geest - Belgium

- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com