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


Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
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
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
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

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

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

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
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

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



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}52n 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 + 261 = 44
219 + 2261 = 444
2109 + 21161 = 4224
21009 + 210161 = 42024
210009 + 2100161 = 420024
2100009 + 21000161 = 4200024
21000009 + 210000161 = 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


    
Palindromic Quasipronics of form n(n+X)
up down
Case X = 0   [Scanned exhaustively upto length 45]
See in-depth webpage about Palindromic Square Numbers
up down
Case X = 1   [Scanned exhaustively upto length 39]
See in-depth webpage about Palindromic Pronic Numbers
up down
Case X = 2   [Scanned exhaustively upto length 47]
See in-depth webpage about Palindromic Quasipronic Numbers
up down
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.
7129.999.999.999.999.999.99820
899.999.999.999.999.999.969.999.999.999.999.999.99839
7029.848.768.829.132.626.52320
890.949.000.614.999.508.181.805.999.416.000.949.09839
699.382.541.482.587.317.00819
88.032.084.672.471.808.733.780.817.427.648.023.08838
682.999.999.999.999.999.99819
8.999.999.999.999.999.996.999.999.999.999.999.99837
67Prime!    2.915.956.745.919.295.16319
8.502.803.744.072.244.896.984.422.704.473.082.05837
662.124.902.219.343.678.98619
4.515.209.441.771.692.447.442.961.771.449.025.15437
65299.999.999.999.999.99818
89.999.999.999.999.999.699.999.999.999.999.99835
64201.273.817.252.896.17618
40.511.149.511.552.246.964.225.511.594.111.50435
63200.007.029.236.835.99618
40.002.811.744.144.569.496.544.144.711.820.00435
6263.981.601.780.397.66117
4.093.645.366.385.385.335.835.836.635.463.90434
6129.999.999.999.999.99817
899.999.999.999.999.969.999.999.999.999.99833
6029.047.072.823.714.84917
843.732.439.626.193.818.391.626.934.237.34833
5920.094.657.512.269.78117
403.795.260.535.420.404.024.535.062.597.30433
589.446.469.351.863.29916
89.235.783.215.692.644.629.651.238.753.29832
579.130.277.032.071.25816
83.361.958.682.367.966.976.328.685.916.33832
566.715.682.777.346.38616
45.100.395.165.946.888.864.956.159.300.15432
552.999.999.999.999.99816
8.999.999.999.999.996.999.999.999.999.99831
54936.880.964.592.24315
877.745.941.815.294.492.518.149.547.77830
53936.259.984.512.71815
876.582.758.599.757.757.995.857.285.67830
52918.039.000.965.24815
842.795.607.293.273.372.392.706.597.24830
51910.127.769.311.06815
828.332.556.471.143.341.174.655.233.82830
50299.999.999.999.99815
89.999.999.999.999.699.999.999.999.99829
49212.353.038.141.39115
45.093.812.807.879.697.870.821.839.05429
4829.999.999.999.99814
899.999.999.999.969.999.999.999.99827
4729.854.146.767.79814
891.270.079.233.313.332.970.072.19827
4628.814.579.197.81414
830.279.974.347.181.743.479.972.03827
452.999.999.999.99813
8.999.999.999.996.999.999.999.99825
442.013.969.414.82113
4.056.072.803.840.483.082.706.50425
432.010.463.667.56113
4.041.964.158.588.858.514.691.40425
42299.999.999.99812
89.999.999.999.699.999.999.99823
41287.446.679.72912
82.625.593.688.088.639.552.62823
40283.162.754.88412
80.181.145.754.345.754.118.10823
39202.055.469.35612
40.826.412.697.279.621.462.80423
38200.952.736.81612
40.382.002.434.443.420.028.30423
3791.222.338.35811
8.321.515.015.775.105.151.23822
3629.999.999.99811
899.999.999.969.999.999.99821
3521.280.530.84611
452.860.993.151.399.068.25421
342.999.999.99810
8.999.999.996.999.999.99819
332.946.920.84410
8.684.342.469.642.434.86819
322.134.473.70610
4.555.978.008.008.795.55419
31299.999.9989
89.999.999.699.999.99817
30294.174.6699
86.538.736.763.783.56817
29294.127.3289
86.510.885.958.801.56817
28289.053.6839
83.552.032.523.025.53817
27212.325.2069
45.081.993.739.918.05417
2629.999.9988
899.999.969.999.99815
2528.862.8838
833.066.101.660.33815
2421.341.6918
455.467.838.764.55415
239.443.4237
89.178.266.287.19814
222.999.9987
8.999.996.999.99813
212.861.4197
8.187.727.277.81813
202.122.2067
4.503.764.673.05413
19636.7766
405.485.584.50412
18299.9986
89.999.699.99811
17Prime!    212.6716
45.229.592.25411
16212.2066
45.032.023.05411
1563.7015
4.058.008.50410
1429.9985
899.969.9989
1329.3695
862.626.2689
1228.8145
830.333.0389
112.9984
8.996.9987
102.1264
4.526.2547
96713
452.2546
82983
89.6985
7Prime!    2113
45.1545
6882
8.0084
5662
4.5544
4282
8683
381
882
211
41
101
01
up down
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.
8514.554.455.445.544.554.45220
211.832.173.316.341.535.101.535.143.613.371.238.11239
848.582.154.816.793.069.36919
73.653.381.299.404.482.100.128.440.499.218.335.63738
838.337.580.895.150.012.45819
69.515.255.183.170.483.066.038.407.138.155.251.59638
822.782.718.512.264.031.30719
7.743.522.318.496.943.766.673.496.948.132.253.47737
81Prime!    749.333.605.100.067.51118
561.500.851.732.263.925.529.362.237.158.005.16536
80710.620.641.519.688.84518
504.981.696.153.854.124.421.458.351.696.189.40536
79501.710.484.827.137.03218
251.713.410.585.480.900.009.084.585.014.317.15236
78282.062.766.593.827.79918
79.559.404.298.564.180.508.146.589.240.495.59735
77229.161.536.782.656.47518
52.515.009.940.588.817.471.888.504.990.051.52535
76145.544.554.455.445.54418
21.183.217.331.634.153.935.143.613.371.238.11235
7585.912.342.960.955.02717
7.380.930.673.040.759.119.570.403.760.390.83734
7428.172.587.468.596.94917
793.694.684.675.745.959.547.576.486.496.39733
7315.914.477.785.786.56417
253.270.603.194.294.080.492.491.306.072.35233
728.326.337.287.030.81816
69.327.892.617.399.755.799.371.629.872.39632
717.119.655.307.820.64516
50.689.491.702.178.711.787.120.719.498.60532
704.893.930.825.891.95416
23.950.558.928.615.522.551.682.985.505.93232
692.633.793.004.698.07816
6.936.865.591.596.540.456.951.955.686.39631
681.584.019.052.709.08216
2.509.116.359.345.383.835.439.536.119.05231
671.455.445.544.554.45216
2.118.321.733.163.411.143.613.371.238.11231
661.107.790.500.876.91316
1.227.199.793.833.126.213.383.979.917.22131
651.099.608.221.780.34316
1.209.138.241.406.932.396.041.428.319.02131
64882.506.981.544.63915
778.818.572.475.033.330.574.275.818.87730
63266.763.018.566.98715
71.162.508.074.971.717.947.080.526.11729
62242.077.790.404.38115
58.601.656.607.068.386.070.665.610.68529
61240.901.135.079.21515
58.033.356.882.455.155.428.865.333.08529
6074.920.057.059.26114
5.613.014.949.763.223.679.494.103.16528
5971.136.750.663.64514
5.060.437.294.981.881.894.927.340.60528
5834.727.947.714.60314
1.206.030.352.468.338.642.530.306.02128
5726.372.438.389.15814
695.505.506.590.040.095.605.505.59627
5622.546.151.203.55114
508.328.934.093.474.390.439.823.80527
5514.554.455.445.54414
211.832.173.316.383.613.371.238.11227
5410.960.402.230.85314
120.130.417.062.131.260.714.031.02127
538.471.752.832.05913
71.770.596.047.533.574.069.507.71726
522.237.919.705.74512
5.008.284.609.370.739.064.828.00525
51164.555.655.72412
27.078.563.831.413.836.587.07223
50145.544.554.45212
21.183.217.331.213.371.238.11223
4927.056.705.66911
732.065.321.767.123.560.23721
4826.267.929.98811
690.004.145.959.541.400.09621
475.215.527.32410
27.201.725.288.252.710.27220
463.495.950.33310
12.221.668.744.786.612.22120
453.476.828.55310
12.088.336.800.863.388.02120
442.718.194.77710
7.388.582.856.582.858.83719
432.241.769.20110
5.025.529.159.519.255.20519
421.667.309.57410
2.779.921.222.221.299.77219
411.583.998.41410
2.509.050.981.890.509.05219
401.455.445.54410
2.118.321.737.371.238.11219
391.106.902.51310
1.225.233.177.713.325.22119
38486.849.8549
237.022.782.287.220.73218
37282.442.3479
79.773.680.508.637.79717
36277.945.1579
77.253.511.411.535.27717
35223.674.4959
50.030.280.608.203.00517
34Prime!    110.651.0639
12.243.658.185.634.22117
3385.559.3278
7.320.398.778.930.23716
3214.554.4528
211.832.131.238.11215
318.308.3887
69.029.344.392.09614
307.395.0917
54.687.400.478.64514
295.040.8827
25.410.511.501.45214
283.504.0837
12.278.611.687.22114
272.705.6697
7.320.655.560.23713
261.659.0227
2.752.360.632.57213
25281.3496
79.158.385.19711
24266.7376
71.149.694.11711
23262.8086
69.069.096.09611
22145.5446
21.183.638.11211
2173.4915
5.401.221.04510
2023.2555
540.888.0459
1911.2235
126.000.6219
1811.0635
122.434.2219
178.4594
71.588.5178
168.3384
69.555.5968
155.4124
29.311.3928
141.5824
2.509.0527
131.4524
2.114.1127
12Prime!    1.1234
1.265.6217
115243
276.6726
10Prime!    2693
73.4375
92353
56.1655
81443
21.3125
7442
2.1124
6332
1.2214
5212
5253
4142
2523
3Prime!    71
772
211
Prime!    51
101
01
up down
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.
5221.101.628.452.340.870.04820
445.278.723.340.641.742.616.247.146.043.327.872.54439
5120.076.590.060.561.403.78720
403.069.468.459.832.951.080.159.238.954.864.960.30439
507.869.324.084.711.774.56419
61.926.261.550.224.808.533.580.842.205.516.262.91638
496.700.829.702.909.744.67219
44.901.118.707.397.497.077.079.479.370.781.110.94438
482.569.010.347.238.204.64419
6.599.814.164.216.960.812.180.696.124.614.189.95637
472.111.446.108.819.027.07219
4.458.204.670.447.010.722.270.107.440.764.028.54437
46251.242.410.444.831.41618
63.122.748.806.129.135.453.192.160.884.722.13635
4568.277.287.222.041.22717
4.661.787.950.401.114.554.111.040.597.871.66434
4425.870.618.469.371.91417
669.288.899.987.807.323.708.789.998.882.96633
4321.854.071.706.057.61717
477.600.450.133.508.191.805.331.054.006.77433
4221.654.046.835.524.84317
468.897.744.355.103.575.301.553.447.798.86433
4120.441.623.013.274.30217
417.859.951.416.825.656.528.614.159.958.71433
402.638.211.511.329.97116
6.960.159.978.513.982.893.158.799.510.69631
392.602.627.310.579.93616
6.773.668.917.776.563.656.777.198.663.77631
382.169.155.049.742.67816
4.705.233.629.824.170.714.289.263.325.07431
37638.147.557.660.60815
407.232.305.348.202.202.843.503.232.70430
36634.272.577.528.88715
402.301.702.605.141.141.506.207.103.20430
35204.040.586.040.40215
41.632.560.751.711.711.715.706.523.61429
3422.104.542.245.33214
488.610.787.875.777.578.787.016.88427
337.989.716.477.39113
63.835.569.389.133.198.396.553.83626
322.546.322.790.44913
6.483.759.753.172.713.579.573.84625
312.471.952.825.15613
6.110.550.769.809.089.670.550.11625
302.228.557.243.29713
4.966.467.386.662.666.837.646.69425
292.048.192.235.51813
4.195.091.433.646.463.341.905.91425
282.008.491.060.20813
4.034.036.338.945.498.336.304.30425
2781.294.635.98911
6.608.817.840.990.487.188.06622
2626.331.005.92411
693.321.873.101.378.123.39621
2522.222.222.22211
493.827.160.595.061.728.39421
248.161.664.18110
66.612.762.244.226.721.66620
23249.281.9869
62.141.509.790.514.12617
22211.313.1489
44.653.247.574.235.64417
2179.585.8918
6.333.914.444.193.33616
206.887.0787
47.431.877.813.47414
192.233.0977
4.986.733.376.89413
182.187.1427
4.783.601.063.87413
172.040.4027
4.163.250.523.61413
16207.0676
42.877.777.82411
1582.5665
6.817.557.18610
1464.4935
4.159.669.51410
1325.1165
630.939.0369
1220.4025
416.343.6149
11Prime!    2.2074
4.881.8847
102.1784
4.754.5747
92.0674
4.282.8247
88143
666.6666
72033
42.2245
62023
41.8145
5242
6963
4182
4143
361
662
211
61
101
01
up down
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.
6414.078.738.935.233.526.22720
198.210.890.006.460.443.878.344.064.600.098.012.89139
634.471.112.010.115.680.89719
19.990.842.607.000.684.622.648.600.070.624.809.99138
622.658.438.045.166.145.29319
7.067.292.839.986.795.976.795.976.899.382.927.60737
612.311.662.135.070.284.53519
5.343.781.826.717.706.435.346.077.176.281.873.43537
60782.618.595.888.768.18218
612.491.866.630.907.042.240.709.036.668.194.21636
59782.345.167.260.793.26218
612.063.960.736.318.589.985.813.637.069.360.21636
58279.573.353.506.965.48318
78.161.259.991.130.690.809.603.119.995.216.18735
57165.225.528.364.325.58818
27.299.475.223.270.560.406.507.232.257.499.27235
5624.430.222.115.159.04917
596.835.752.596.006.424.600.695.257.538.69533
5515.870.259.146.035.62817
251.865.125.362.327.595.723.263.521.568.15233
5413.999.785.814.483.85717
195.994.002.851.423.515.324.158.200.499.59133
537.325.505.333.667.73516
53.663.028.393.594.477.449.539.382.036.63532
522.762.947.364.366.12116
7.633.878.138.257.711.177.528.318.783.36731
512.663.237.144.235.05116
7.092.832.086.433.285.823.346.802.382.90731
501.667.449.597.470.60616
2.780.388.160.104.895.984.010.618.830.87231
491.391.554.098.835.91716
1.936.422.809.987.049.407.899.082.246.39131
48716.255.983.645.01515
513.022.634.107.292.292.701.436.220.31530
47525.758.362.123.45615
276.421.855.342.742.247.243.558.124.67230
46440.309.354.779.71715
193.872.327.906.533.335.609.723.278.39130
45279.399.473.798.43315
78.064.065.958.842.924.885.956.046.08729
4485.263.388.929.19114
7.269.845.491.691.001.961.945.489.62728
4326.528.700.343.14314
703.771.941.896.434.698.149.177.30727
4215.977.390.443.07814
255.277.005.370.656.073.500.772.55227
4115.849.612.021.11614
251.210.201.219.999.912.102.012.15227
401.656.680.099.18813
2.744.588.951.055.501.598.854.47225
391.389.555.242.01713
1.930.863.770.625.260.773.680.39125
38849.022.995.99112
720.840.047.726.627.740.048.02724
37436.181.402.10712
190.254.215.546.645.512.452.09124
36247.041.361.04212
61.029.434.066.966.043.492.01623
3588.440.105.03311
7.821.652.178.778.712.561.28722
3424.698.565.24211
610.019.125.161.521.910.01621
337.815.370.55210
61.080.016.911.961.008.01620
322.352.019.43910
5.531.995.455.545.991.35519
311.414.071.39710
1.999.597.924.297.959.99119
30782.357.2629
612.082.890.098.280.21618
29440.021.4179
193.618.850.058.816.39118
28272.515.5139
74.264.706.460.746.24717
27235.769.7559
55.587.378.787.378.55517
26235.680.7559
55.545.419.691.454.55517
2588.837.6118
7.892.121.661.212.98716
2484.890.5038
7.206.398.008.936.02716
2373.114.0358
5.345.662.552.665.43516
2223.975.4758
574.823.545.328.47515
2123.887.4198
570.608.929.806.07515
2014.529.4868
211.106.050.601.11215
197.560.0697
57.154.688.645.17514
187.153.6797
51.175.166.157.11514
175.401.3967
29.175.111.157.19214
162.436.0997
5.934.592.954.39513
152.273.9157
5.170.703.070.71513
141.731.7467
2.998.954.598.99213
131.378.5077
1.900.289.820.09113
12877.1736
769.437.734.96712
11486.6186
236.799.997.63212
108.5414
72.999.9278
9Prime!    2.7314
7.474.7477
8Prime!    8633
749.9476
77153
515.5156
62733
76.1675
5Prime!    1373
19.5915
4222
6163
3Prime!    51
552
211
Prime!    71
101
01
up down
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.
9728.736.854.321.862.795.05620
825.806.796.315.964.402.909.204.469.513.697.608.52839
9622.304.559.111.550.001.83920
497.493.357.160.628.207.535.702.826.061.753.394.79439
9522.183.289.860.875.017.85420
492.098.349.051.600.369.131.963.006.150.943.890.29439
9421.109.211.394.290.929.32920
445.598.805.688.862.000.797.000.268.886.508.895.54439
9321.000.000.000.000.000.00920
441.000.000.000.000.000.525.000.000.000.000.000.14439
9220.990.098.888.890.099.00420
440.584.251.365.385.368.919.863.583.563.152.485.04439
919.000.000.000.000.000.00219
81.000.000.000.000.000.099.000.000.000.000.000.01838
902.940.416.984.997.499.91619
8.646.052.045.661.787.666.667.871.665.402.506.46837
892.110.423.522.968.524.90419
4.453.887.446.298.879.977.799.788.926.447.883.54437
882.100.000.000.000.000.00919
4.410.000.000.000.000.052.500.000.000.000.000.14437
872.099.009.888.890.099.00419
4.405.842.513.658.425.780.875.248.563.152.485.04437
862.098.890.099.009.889.00419
4.405.339.647.721.741.680.861.471.277.469.335.04437
85900.000.000.000.000.00218
810.000.000.000.000.009.900.000.000.000.000.01836
84210.000.000.000.000.00918
44.100.000.000.000.005.250.000.000.000.000.14435
8390.000.000.000.000.00217
8.100.000.000.000.000.990.000.000.000.000.01834
8266.891.629.092.892.82417
4.474.490.042.701.146.116.411.072.400.944.74434
8121.000.000.000.000.00917
441.000.000.000.000.525.000.000.000.000.14433
809.000.000.000.000.00216
81.000.000.000.000.099.000.000.000.000.01832
792.219.536.203.158.25416
4.926.340.957.130.173.710.317.590.436.29431
782.100.000.000.000.00916
4.410.000.000.000.052.500.000.000.000.14431
772.099.198.088.021.02916
4.406.632.612.751.158.511.572.162.366.04431
762.099.009.780.099.00416
4.405.842.056.951.283.821.596.502.485.04431
75945.318.054.069.68615
893.626.223.350.104.401.053.322.626.39830
74900.000.000.000.00215
810.000.000.000.009.900.000.000.000.01830
73895.895.199.033.81715
802.628.207.651.848.848.156.702.826.20830
72667.990.192.880.10915
446.210.897.784.009.900.487.798.012.64430
71299.692.501.995.80715
89.815.595.752.508.880.525.759.551.89829
70295.580.359.679.16215
87.367.749.028.064.846.082.094.776.37829
69294.572.920.786.61615
86.773.205.660.760.006.706.650.237.76829
68284.366.947.570.41715
80.864.560.870.518.281.507.806.546.80829
67222.266.487.782.09915
49.402.391.590.991.519.909.519.320.49429
66210.000.000.000.00915
44.100.000.000.005.250.000.000.000.14429
6590.000.000.000.00214
8.100.000.000.000.990.000.000.000.01828
6422.231.909.422.37414
494.257.796.564.797.465.697.752.49427
6322.199.010.990.21414
492.796.088.943.797.349.880.697.29427
6221.000.000.000.00914
441.000.000.000.525.000.000.000.14427
619.000.000.000.00213
81.000.000.000.099.000.000.000.01826
606.673.701.200.52913
44.538.287.713.988.931.778.283.54426
592.948.655.661.41613
8.694.570.209.621.269.020.754.96825
582.221.874.942.95913
4.936.728.262.164.612.628.276.39425
572.220.153.281.41413
4.929.080.592.988.892.950.809.29425
562.219.890.109.77913
4.927.912.099.510.159.902.197.29425
552.100.000.000.00913
4.410.000.000.052.500.000.000.14425
54900.000.000.00212
810.000.000.009.900.000.000.01824
53291.222.254.25712
84.810.401.376.567.310.401.84823
52284.038.530.77612
80.677.886.967.376.968.877.60823
51210.000.000.00912
44.100.000.005.250.000.000.14423
50209.902.199.00412
44.058.933.148.184.133.985.04423
4992.917.907.01611
8.633.737.444.884.447.373.36822
4890.000.000.00211
8.100.000.000.990.000.000.01822
47Prime!    29.097.613.05711
846.671.085.818.580.176.64821
4622.199.009.77911
492.796.035.323.530.697.29421
4521.000.000.00911
441.000.000.525.000.000.14421
449.000.000.00210
81.000.000.099.000.000.01820
432.984.477.00710
8.907.103.026.203.017.09819
422.219.890.21410
4.927.912.577.752.197.29419
412.100.000.00910
4.410.000.052.500.000.14419
40900.000.0029
810.000.009.900.000.01818
39703.210.9749
494.505.678.876.505.49418
38665.012.0449
442.241.023.320.142.24418
37221.990.2149
49.279.656.665.697.29417
36210.000.0099
44.100.005.250.000.14417
3590.000.0028
8.100.000.990.000.01816
3429.857.8868
891.493.565.394.19815
3329.802.8978
888.212.878.212.88815
3228.867.7228
833.345.575.543.33815
3121.155.3448
447.548.727.845.74415
3021.152.8248
447.442.111.244.74415
2921.117.3048
445.940.676.049.54415
2821.037.9248
442.594.393.495.24415
2721.000.0098
441.000.525.000.14415
269.000.0027
81.000.099.000.01814
258.972.8767
80.512.566.521.50814
242.219.7797
4.927.434.347.29413
232.109.6647
4.450.696.960.54413
222.100.0097
4.410.052.500.14413
21942.9976
889.249.942.98812
20900.0026
810.009.900.01812
19669.0246
447.597.795.74412
18292.9676
85.831.713.85811
17223.1146
49.781.418.79411
16210.0096
44.105.250.14411
1590.0025
8.100.990.01810
1429.6965
882.060.2889
1321.0095
441.525.1449
129.0024
81.099.0188
112.9824
8.913.1987
102.9724
8.853.5887
92.1094
4.462.6447
89023
819.9186
76643
445.5446
62193
49.4945
5262
8583
4Prime!    192
4943
341
442
211
81
101
01
up down
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.
10823.683.958.404.363.815.50520
560.929.885.699.635.409.979.904.536.996.588.929.06539
10722.133.746.889.973.981.09620
489.902.751.389.432.880.606.088.234.983.157.209.98439
10619.890.991.099.890.991.09920
395.651.526.935.942.619.999.916.249.539.625.156.59339
1056.999.999.999.999.999.99619
48.999.999.999.999.999.999.999.999.999.999.999.98438
1042.412.063.028.201.142.84719
5.818.048.052.014.867.252.527.684.102.508.408.18537
1031.989.000.119.790.001.09919
3.956.121.476.524.638.737.378.364.256.741.216.59337
102699.999.999.999.999.99618
489.999.999.999.999.999.999.999.999.999.999.98436
101294.011.283.367.553.97418
86.442.634.747.436.122.422.163.474.743.624.46835
100238.427.502.869.428.60518
56.847.674.124.551.387.578.315.542.147.674.86535
99Prime!    183.952.041.061.237.03918
33.838.353.410.595.039.093.059.501.435.383.83335
9874.972.390.516.353.90517
5.620.859.339.736.673.553.766.379.339.580.26534
9769.999.999.999.999.99617
4.899.999.999.999.999.999.999.999.999.999.98434
9629.440.591.428.026.31817
866.748.423.631.976.949.679.136.324.847.66833
9524.197.921.094.983.44517
585.539.385.319.044.999.440.913.583.935.58533
9423.672.512.354.237.73717
560.387.841.161.538.474.835.161.148.783.06533
939.108.243.842.099.28416
82.960.105.887.139.599.593.178.850.106.92832
926.999.999.999.999.99616
48.999.999.999.999.999.999.999.999.999.98432
913.004.385.430.296.78116
9.026.331.813.779.597.959.773.181.336.20931
902.913.451.145.839.30416
8.488.197.579.192.376.732.919.757.918.84831
891.835.437.923.830.08916
3.368.832.372.233.722.273.322.732.388.63331
88951.856.579.421.42115
906.030.947.787.855.558.787.749.030.60930
87930.589.355.492.66815
865.996.548.556.266.662.655.845.699.56830
86699.999.999.999.99615
489.999.999.999.999.999.999.999.999.98430
85697.090.484.234.13615
485.935.143.209.787.787.902.341.539.58430
84630.600.416.802.79915
397.656.885.671.868.868.176.588.656.79330
83630.477.098.800.54915
397.501.372.111.962.269.111.273.105.79330
82625.328.055.612.04315
391.035.177.135.543.345.531.771.530.19330
81294.411.411.576.17415
86.678.079.266.277.677.266.297.087.66829
80225.762.237.924.33515
50.968.588.072.605.850.627.088.586.90529
79220.050.252.330.34615
48.422.113.550.650.705.605.531.122.48429
78183.014.335.582.88915
33.494.247.028.847.774.882.074.249.43329
7776.761.240.619.74514
5.892.288.061.483.003.841.608.822.98528
7671.108.945.208.78514
5.056.482.088.706.556.078.802.846.50528
7569.999.999.999.99614
4.899.999.999.999.999.999.999.999.98428
7469.696.500.774.73614
4.857.602.220.243.333.420.222.067.58428
7363.220.385.270.74914
3.996.817.113.782.442.873.117.186.99328
7258.198.234.728.95314
3.387.034.525.566.776.655.254.307.83328
7155.827.247.193.61914
3.116.681.529.217.887.129.251.866.11328
7022.388.140.098.98514
501.228.817.091.959.190.718.822.10527
6918.956.175.005.05914
359.336.570.822.575.228.075.633.95327
686.999.999.999.99613
48.999.999.999.999.999.999.999.98426
676.933.240.247.18613
48.069.820.325.255.252.302.896.08426
663.010.857.767.52113
9.065.264.496.265.626.944.625.60925
651.830.169.111.63913
3.349.518.977.212.127.798.159.43325
64906.998.967.65812
822.647.127.339.933.721.746.22824
63722.945.524.91712
522.650.232.003.300.232.056.22524
62714.048.799.58512
509.865.688.194.491.886.568.90524
61699.999.999.99612
489.999.999.999.999.999.999.98424
60228.640.304.57512
52.276.388.877.977.888.367.22523
59197.920.018.79312
39.172.333.840.604.833.327.19323
5893.910.205.19411
8.819.126.640.330.466.219.18822
5793.236.732.76811
8.693.088.337.997.338.803.96822
5669.999.999.99611
4.899.999.999.999.999.999.98422
5562.884.810.89311
3.954.499.441.551.449.944.59322
54Prime!    18.963.885.55911
359.628.955.646.559.826.95321
537.228.936.82510
52.257.527.677.672.575.22520
526.999.999.99610
48.999.999.999.999.999.98420
511.771.260.81910
3.137.364.903.094.637.31319
50699.999.9969
489.999.999.999.999.98418
49695.334.3769
483.489.900.009.984.38418
48287.876.0589
82.872.627.072.627.82817
47225.207.1859
50.718.277.977.281.70517
46182.901.1039
33.452.814.941.825.43317
4591.842.9388
8.435.125.995.215.34816
4491.050.7848
8.290.245.995.420.92816
4369.999.9968
4.899.999.999.999.98416
4269.692.7368
4.857.078.008.707.58416
4159.318.0098
3.518.626.666.268.15316
4057.524.7538
3.309.097.667.909.03316
3923.352.3278
545.331.363.133.54515
38Prime!    19.782.1998
391.335.555.533.19315
37Prime!    17.869.1698
319.307.343.703.91315
3617.646.6198
311.403.303.304.11315
357.243.2257
52.464.366.346.42514
346.999.9967
48.999.999.999.98414
332.281.8177
5.206.707.076.02513
32Prime Curios!    1.934.0637
3.740.615.160.47313
31896.4786
803.679.976.30812
30699.9966
489.999.999.98412
29597.8836
357.468.864.75312
28298.1446
88.892.229.88811
27283.7786
80.532.223.50811
26241.0976
58.129.692.18511
25228.1756
52.065.656.02511
24188.5836
35.565.056.55311
2393.8445
8.807.447.08810
2269.9965
4.899.999.98410
2157.4895
3.305.445.03310
2030.0315
902.101.2099
1928.3145
801.909.1089
189.3944
88.322.3888
176.9964
48.999.9848
162.8284
8.020.2087
152.2354
5.013.1057
149243
861.1686
137073
505.5056
126963
489.9846
112943
88.7885
102843
82.9285
92253
52.4255
82163
48.3845
7Prime!    1733
31.3135
6912
9.0094
5882
8.4484
4662
4.8844
3Prime!    31
332
211
91
101
01
up down
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.
7625.210.573.274.064.841.32820
635.573.004.806.992.453.404.354.299.608.400.375.53639
7516.681.463.061.790.801.32720
278.271.209.881.890.936.121.639.098.188.902.172.87239
7415.620.041.971.084.230.44220
243.985.711.178.432.931.060.139.234.871.117.589.34239
7315.445.446.545.435.545.54420
238.561.818.987.906.827.999.728.609.789.818.165.83239
7215.444.456.545.434.555.54420
238.531.237.983.816.285.595.582.618.389.732.135.83239
7115.219.869.175.229.085.44420
231.644.417.711.088.481.737.184.880.117.714.446.13239
701.696.782.851.936.705.26919
2.879.072.046.626.459.090.909.546.266.402.709.78237
691.684.230.962.345.308.79719
2.836.633.934.522.604.993.994.062.254.393.366.38237
681.668.295.293.997.506.03919
2.783.209.187.974.225.124.215.224.797.819.023.87237
671.599.655.235.294.309.48719
2.558.896.871.804.492.662.662.944.081.786.988.55237
661.438.728.178.155.345.52919
2.069.938.770.618.199.676.769.918.160.778.399.60237
65262.139.313.895.938.91318
68.717.019.889.833.594.449.533.898.891.071.78635
64251.096.226.342.076.22818
63.049.314.883.231.178.087.113.238.841.394.03635
63172.610.504.583.658.59218
29.794.386.292.625.225.352.252.629.268.349.79235
6250.853.897.306.350.40417
2.586.118.871.244.833.333.384.421.788.116.85234
6148.313.653.281.364.57217
2.334.209.093.391.910.110.191.933.909.024.33234
6045.108.988.816.777.75417
2.034.820.872.072.180.880.812.702.780.284.30234
5925.240.619.955.431.64817
637.088.895.734.534.555.435.437.598.880.73633
5816.517.616.586.567.12717
272.831.657.700.837.616.738.007.756.138.27233
571.545.039.109.604.07216
2.387.145.850.206.157.516.020.585.417.83231
561.510.415.941.428.60216
2.281.356.316.121.663.661.216.136.531.82231
55458.433.179.591.05915
210.160.980.149.972.279.941.089.061.01230
54165.822.864.864.83915
27.497.222.511.984.148.911.522.279.47229
53153.465.444.564.34715
23.551.642.675.334.043.357.624.615.53229
52147.647.028.000.83215
21.799.644.877.479.797.477.844.699.71229
5152.409.339.938.52714
2.746.738.912.792.552.972.198.376.47228
5050.840.261.816.21214
2.584.732.221.541.441.451.222.374.85228
4916.828.270.461.92214
283.190.686.739.747.937.686.091.38227
484.855.892.951.01913
23.579.696.351.799.715.369.697.53226
474.515.673.445.36213
20.391.306.665.188.156.660.319.30226
462.514.764.874.26813
6.324.042.372.874.782.732.404.23625
45544.332.666.72412
296.298.052.067.760.250.892.69224
44262.573.714.11812
68.944.955.348.084.355.944.98623
43152.730.610.57212
23.326.639.407.070.493.662.33223
4247.972.497.14411
2.301.360.482.662.840.631.03222
4116.700.003.32711
278.890.111.272.111.098.87221
4015.114.678.51411
228.453.506.717.605.354.82221
391.665.597.36410
2.774.214.593.954.124.77219
381.632.565.35710
2.665.269.659.569.625.66219
37Prime!    1.534.664.34710
2.355.194.671.764.915.53219
361.458.904.65910
2.128.402.817.182.048.21219
351.417.320.72910
2.008.798.061.608.978.00219
34514.668.9579
264.884.139.931.488.46218
33497.665.8749
247.671.326.623.176.74218
32164.399.7279
27.027.271.717.272.07217
31151.698.4729
23.012.427.772.421.03217
3054.072.5248
2.923.838.338.383.29216
2952.030.3528
2.707.157.997.517.07216
2849.407.0178
2.441.053.773.501.44216
2748.675.3198
2.369.287.117.829.63216
2626.216.7538
687.318.373.813.78615
2526.211.4388
687.039.717.930.78615
2426.132.5788
682.911.868.119.28615
2325.234.0838
636.759.171.957.63615
2215.336.6448
235.212.787.212.53215
2115.019.6148
225.588.939.885.52215
2014.790.5328
218.759.969.957.81215
1914.674.1848
215.331.808.133.51215
181.547.1477
2.393.677.763.93213
17487.6196
237.776.677.73212
16160.1876
25.661.316.65211
1516.9975
289.050.9829
1415.9045
253.080.3529
135.3924
29.122.1928
121.6944
2.884.8827
111.6644
2.783.8727
101.6394
2.701.0727
92583
68.8865
82483
63.7365
7Prime!    1673
29.3925
6Prime!    1573
26.0625
5Prime!    1373
20.0025
4442
2.3324
3122
2523
2Prime!    21
222
101
01




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 n2Y.
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 n2Y.
whereby Y = (X/2)2 or the square of the natural numbers.

n(n+0) = n20 (trivial case)
n(n+2) = n21
n(n+4) = n24
n(n+6) = n29
n(n+8) = n216
n(n+10) = n225

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 resumée :

%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.


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( © All rights reserved ) - Last modified : August 21, 2021.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com