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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 )


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
[84] 20 99009888890099 004
[85] 20 98890099009889 004
[90] 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  [71]
 1544465454355544  [72]
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 34]
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.
7029.999.999.999.999.999.99820
899.999.999.999.999.999.969.999.999.999.999.999.99839
6929.848.768.829.132.626.52320
890.949.000.614.999.508.181.805.999.416.000.949.09839
689.382.541.482.587.317.00819
88.032.084.672.471.808.733.780.817.427.648.023.08838
672.999.999.999.999.999.99819
8.999.999.999.999.999.996.999.999.999.999.999.99837
66Prime!    2.915.956.745.919.295.16319
8.502.803.744.072.244.896.984.422.704.473.082.05837
652.124.902.219.343.678.98619
4.515.209.441.771.692.447.442.961.771.449.025.15437
64299.999.999.999.999.99818
89.999.999.999.999.999.699.999.999.999.999.99835
63201.273.817.252.896.17618
40.511.149.511.552.246.964.225.511.594.111.50435
62200.007.029.236.835.99618
40.002.811.744.144.569.496.544.144.711.820.00435
6163.981.601.780.397.66117
4.093.645.366.385.385.335.835.836.635.463.90434
6029.999.999.999.999.99817
899.999.999.999.999.969.999.999.999.999.99833
5929.047.072.823.714.84917
843.732.439.626.193.818.391.626.934.237.34833
5820.094.657.512.269.78117
403.795.260.535.420.404.024.535.062.597.30433
579.446.469.351.863.29916
89.235.783.215.692.644.629.651.238.753.29832
569.130.277.032.071.25816
83.361.958.682.367.966.976.328.685.916.33832
556.715.682.777.346.38616
45.100.395.165.946.888.864.956.159.300.15432
542.999.999.999.999.99816
8.999.999.999.999.996.999.999.999.999.99831
53936.880.964.592.24315
877.745.941.815.294.492.518.149.547.77830
52936.259.984.512.71815
876.582.758.599.757.757.995.857.285.67830
51918.039.000.965.24815
842.795.607.293.273.372.392.706.597.24830
50910.127.769.311.06815
828.332.556.471.143.341.174.655.233.82830
49299.999.999.999.99815
89.999.999.999.999.699.999.999.999.99829
48212.353.038.141.39115
45.093.812.807.879.697.870.821.839.05429
4729.999.999.999.99814
899.999.999.999.969.999.999.999.99827
4629.854.146.767.79814
891.270.079.233.313.332.970.072.19827
4528.814.579.197.81414
830.279.974.347.181.743.479.972.03827
442.999.999.999.99813
8.999.999.999.996.999.999.999.99825
432.013.969.414.82113
4.056.072.803.840.483.082.706.50425
422.010.463.667.56113
4.041.964.158.588.858.514.691.40425
41299.999.999.99812
89.999.999.999.699.999.999.99823
40287.446.679.72912
82.625.593.688.088.639.552.62823
39283.162.754.88412
80.181.145.754.345.754.118.10823
38202.055.469.35612
40.826.412.697.279.621.462.80423
37200.952.736.81612
40.382.002.434.443.420.028.30423
3691.222.338.35811
8.321.515.015.775.105.151.23822
3529.999.999.99811
899.999.999.969.999.999.99821
3421.280.530.84611
452.860.993.151.399.068.25421
332.999.999.99810
8.999.999.996.999.999.99819
322.946.920.84410
8.684.342.469.642.434.86819
312.134.473.70610
4.555.978.008.008.795.55419
30299.999.9989
89.999.999.699.999.99817
29294.174.6699
86.538.736.763.783.56817
28294.127.3289
86.510.885.958.801.56817
27289.053.6839
83.552.032.523.025.53817
26212.325.2069
45.081.993.739.918.05417
2529.999.9988
899.999.969.999.99815
2428.862.8838
833.066.101.660.33815
2321.341.6918
455.467.838.764.55415
229.443.4237
89.178.266.287.19814
212.999.9987
8.999.996.999.99813
202.861.4197
8.187.727.277.81813
192.122.2067
4.503.764.673.05413
18636.7766
405.485.584.50412
17299.9986
89.999.699.99811
16Prime!    212.6716
45.229.592.25411
15212.2066
45.032.023.05411
1463.7015
4.058.008.50410
1329.9985
899.969.9989
1229.3695
862.626.2689
1128.8145
830.333.0389
102.9984
8.996.9987
92.1264
4.526.2547
86713
452.2546
72983
89.6985
6Prime!    2113
45.1545
5882
8.0084
4662
4.5544
3282
8683
281
882
111
41
001
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.
8414.554.455.445.544.554.45220
211.832.173.316.341.535.101.535.143.613.371.238.11239
838.582.154.816.793.069.36919
73.653.381.299.404.482.100.128.440.499.218.335.63738
828.337.580.895.150.012.45819
69.515.255.183.170.483.066.038.407.138.155.251.59638
812.782.718.512.264.031.30719
7.743.522.318.496.943.766.673.496.948.132.253.47737
80Prime!    749.333.605.100.067.51118
561.500.851.732.263.925.529.362.237.158.005.16536
79710.620.641.519.688.84518
504.981.696.153.854.124.421.458.351.696.189.40536
78501.710.484.827.137.03218
251.713.410.585.480.900.009.084.585.014.317.15236
77282.062.766.593.827.79918
79.559.404.298.564.180.508.146.589.240.495.59735
76229.161.536.782.656.47518
52.515.009.940.588.817.471.888.504.990.051.52535
75145.544.554.455.445.54418
21.183.217.331.634.153.935.143.613.371.238.11235
7485.912.342.960.955.02717
7.380.930.673.040.759.119.570.403.760.390.83734
7328.172.587.468.596.94917
793.694.684.675.745.959.547.576.486.496.39733
7215.914.477.785.786.56417
253.270.603.194.294.080.492.491.306.072.35233
718.326.337.287.030.81816
69.327.892.617.399.755.799.371.629.872.39632
707.119.655.307.820.64516
50.689.491.702.178.711.787.120.719.498.60532
694.893.930.825.891.95416
23.950.558.928.615.522.551.682.985.505.93232
682.633.793.004.698.07816
6.936.865.591.596.540.456.951.955.686.39631
671.584.019.052.709.08216
2.509.116.359.345.383.835.439.536.119.05231
661.455.445.544.554.45216
2.118.321.733.163.411.143.613.371.238.11231
651.107.790.500.876.91316
1.227.199.793.833.126.213.383.979.917.22131
641.099.608.221.780.34316
1.209.138.241.406.932.396.041.428.319.02131
63882.506.981.544.63915
778.818.572.475.033.330.574.275.818.87730
62266.763.018.566.98715
71.162.508.074.971.717.947.080.526.11729
61242.077.790.404.38115
58.601.656.607.068.386.070.665.610.68529
60240.901.135.079.21515
58.033.356.882.455.155.428.865.333.08529
5974.920.057.059.26114
5.613.014.949.763.223.679.494.103.16528
5871.136.750.663.64514
5.060.437.294.981.881.894.927.340.60528
5734.727.947.714.60314
1.206.030.352.468.338.642.530.306.02128
5626.372.438.389.15814
695.505.506.590.040.095.605.505.59627
5522.546.151.203.55114
508.328.934.093.474.390.439.823.80527
5414.554.455.445.54414
211.832.173.316.383.613.371.238.11227
5310.960.402.230.85314
120.130.417.062.131.260.714.031.02127
528.471.752.832.05913
71.770.596.047.533.574.069.507.71726
512.237.919.705.74512
5.008.284.609.370.739.064.828.00525
50164.555.655.72412
27.078.563.831.413.836.587.07223
49145.544.554.45212
21.183.217.331.213.371.238.11223
4827.056.705.66911
732.065.321.767.123.560.23721
4726.267.929.98811
690.004.145.959.541.400.09621
465.215.527.32410
27.201.725.288.252.710.27220
453.495.950.33310
12.221.668.744.786.612.22120
443.476.828.55310
12.088.336.800.863.388.02120
432.718.194.77710
7.388.582.856.582.858.83719
422.241.769.20110
5.025.529.159.519.255.20519
411.667.309.57410
2.779.921.222.221.299.77219
401.583.998.41410
2.509.050.981.890.509.05219
391.455.445.54410
2.118.321.737.371.238.11219
381.106.902.51310
1.225.233.177.713.325.22119
37486.849.8549
237.022.782.287.220.73218
36282.442.3479
79.773.680.508.637.79717
35277.945.1579
77.253.511.411.535.27717
34223.674.4959
50.030.280.608.203.00517
33Prime!    110.651.0639
12.243.658.185.634.22117
3285.559.3278
7.320.398.778.930.23716
3114.554.4528
211.832.131.238.11215
308.308.3887
69.029.344.392.09614
297.395.0917
54.687.400.478.64514
285.040.8827
25.410.511.501.45214
273.504.0837
12.278.611.687.22114
262.705.6697
7.320.655.560.23713
251.659.0227
2.752.360.632.57213
24281.3496
79.158.385.19711
23266.7376
71.149.694.11711
22262.8086
69.069.096.09611
21145.5446
21.183.638.11211
2073.4915
5.401.221.04510
1923.2555
540.888.0459
1811.2235
126.000.6219
1711.0635
122.434.2219
168.4594
71.588.5178
158.3384
69.555.5968
145.4124
29.311.3928
131.5824
2.509.0527
121.4524
2.114.1127
11Prime!    1.1234
1.265.6217
105243
276.6726
9Prime!    2693
73.4375
82353
56.1655
71443
21.3125
6442
2.1124
5332
1.2214
4212
5253
3142
2523
2Prime!    71
772
111
51
001
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.
5121.101.628.452.340.870.04820
445.278.723.340.641.742.616.247.146.043.327.872.54439
5020.076.590.060.561.403.78720
403.069.468.459.832.951.080.159.238.954.864.960.30439
497.869.324.084.711.774.56419
61.926.261.550.224.808.533.580.842.205.516.262.91638
486.700.829.702.909.744.67219
44.901.118.707.397.497.077.079.479.370.781.110.94438
472.569.010.347.238.204.64419
6.599.814.164.216.960.812.180.696.124.614.189.95637
462.111.446.108.819.027.07219
4.458.204.670.447.010.722.270.107.440.764.028.54437
45251.242.410.444.831.41618
63.122.748.806.129.135.453.192.160.884.722.13635
4468.277.287.222.041.22717
4.661.787.950.401.114.554.111.040.597.871.66434
4325.870.618.469.371.91417
669.288.899.987.807.323.708.789.998.882.96633
4221.854.071.706.057.61717
477.600.450.133.508.191.805.331.054.006.77433
4121.654.046.835.524.84317
468.897.744.355.103.575.301.553.447.798.86433
4020.441.623.013.274.30217
417.859.951.416.825.656.528.614.159.958.71433
392.638.211.511.329.97116
6.960.159.978.513.982.893.158.799.510.69631
382.602.627.310.579.93616
6.773.668.917.776.563.656.777.198.663.77631
372.169.155.049.742.67816
4.705.233.629.824.170.714.289.263.325.07431
36638.147.557.660.60815
407.232.305.348.202.202.843.503.232.70430
35634.272.577.528.88715
402.301.702.605.141.141.506.207.103.20430
34204.040.586.040.40215
41.632.560.751.711.711.715.706.523.61429
3322.104.542.245.33214
488.610.787.875.777.578.787.016.88427
327.989.716.477.39113
63.835.569.389.133.198.396.553.83626
312.546.322.790.44913
6.483.759.753.172.713.579.573.84625
302.471.952.825.15613
6.110.550.769.809.089.670.550.11625
292.228.557.243.29713
4.966.467.386.662.666.837.646.69425
282.048.192.235.51813
4.195.091.433.646.463.341.905.91425
272.008.491.060.20813
4.034.036.338.945.498.336.304.30425
2681.294.635.98911
6.608.817.840.990.487.188.06622
2526.331.005.92411
693.321.873.101.378.123.39621
2422.222.222.22211
493.827.160.595.061.728.39421
238.161.664.18110
66.612.762.244.226.721.66620
22249.281.9869
62.141.509.790.514.12617
21211.313.1489
44.653.247.574.235.64417
2079.585.8918
6.333.914.444.193.33616
196.887.0787
47.431.877.813.47414
182.233.0977
4.986.733.376.89413
172.187.1427
4.783.601.063.87413
162.040.4027
4.163.250.523.61413
15207.0676
42.877.777.82411
1482.5665
6.817.557.18610
1364.4935
4.159.669.51410
1225.1165
630.939.0399
1120.4025
416.343.6149
10Prime!    2.2074
4.881.8847
92.1784
4.754.5747
82.0674
4.282.8247
78143
666.6666
62033
42.2245
52023
41.8145
4242
6963
3182
4143
261
662
111
61
001
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.
6314.078.738.935.233.526.22720
198.210.890.006.460.443.878.344.064.600.098.012.89139
624.471.112.010.115.680.89719
19.990.842.607.000.684.622.648.600.070.624.809.99138
612.658.438.045.166.145.29319
7.067.292.839.986.795.976.795.976.899.382.927.60737
602.311.662.135.070.284.53519
5.343.781.826.717.706.435.346.077.176.281.873.43537
59782.618.595.888.768.18218
612.491.866.630.907.042.240.709.036.668.194.21636
58782.345.167.260.793.26218
612.063.960.736.318.589.985.813.637.069.360.21636
57279.573.353.506.965.48318
78.161.259.991.130.690.809.603.119.995.216.18735
56165.225.528.364.325.58818
27.299.475.223.270.560.406.507.232.257.499.27235
5524.430.222.115.159.04917
596.835.752.596.006.424.600.695.257.538.69533
5415.870.259.146.035.62817
251.865.125.362.327.595.723.263.521.568.15233
5313.999.785.814.483.85717
195.994.002.851.423.515.324.158.200.499.59133
527.325.505.333.667.73516
53.663.028.393.594.477.449.539.382.036.63532
512.762.947.364.366.12116
7.633.878.138.257.711.177.528.318.783.36731
502.663.237.144.235.05116
7.092.832.086.433.285.823.346.802.382.90731
491.667.449.597.470.60616
2.780.388.160.104.895.984.010.618.830.87231
481.391.554.098.835.91716
1.936.422.809.987.049.407.899.082.246.39131
47716.255.983.645.01515
513.022.634.107.292.292.701.436.220.31530
46525.758.362.123.45615
276.421.855.342.742.247.243.558.124.67230
45440.309.354.779.71715
193.872.327.906.533.335.609.723.278.39130
44279.399.473.798.43315
78.064.065.958.842.924.885.956.046.08729
4385.263.388.929.19114
7.269.845.491.691.001.961.945.489.62728
4226.528.700.343.14314
703.771.941.896.434.698.149.177.30727
4115.977.390.443.07814
255.277.005.370.656.073.500.772.55227
4015.849.612.021.11614
251.210.201.219.999.912.102.012.15227
391.656.680.099.18813
2.744.588.951.055.501.598.854.47225
381.389.555.242.01713
1.930.863.770.625.260.773.680.39125
37849.022.995.99112
720.840.047.726.627.740.048.02724
36436.181.402.10712
190.254.215.546.645.512.452.09124
35247.041.361.04212
61.029.434.066.966.043.492.01623
3488.440.105.03311
7.821.652.178.778.712.561.28722
3324.698.565.24211
610.019.125.161.521.910.01621
327.815.370.55210
61.080.016.911.961.008.01620
312.352.019.43910
5.531.995.455.545.991.35519
301.414.071.39710
1.999.597.924.297.959.99119
29782.357.2629
612.082.890.098.280.21618
28440.021.4179
193.618.850.058.816.39118
27272.515.5139
74.264.706.460.746.24717
26235.769.7559
55.587.378.787.378.55517
25235.680.7559
55.545.419.691.454.55517
2488.837.6118
7.892.121.661.212.98716
2384.890.5038
7.206.398.008.936.02716
2273.114.0358
5.345.662.552.665.43516
2123.975.4758
574.823.545.328.47515
2023.887.4198
570.608.929.806.07515
1914.529.4868
211.106.050.601.11215
187.560.0697
57.154.688.645.17514
177.153.6797
51.175.166.157.11514
165.401.3967
29.175.111.157.19214
152.436.0997
5.934.592.954.39513
142.273.9157
5.170.703.070.71513
131.731.7467
2.998.954.598.99213
121.378.5077
1.900.289.820.09113
11877.1736
769.437.734.96712
10486.6186
236.799.997.63212
98.5414
72.999.9278
8Prime!    2.7314
7.474.7477
7Prime!    8633
749.9476
67153
515.5156
52733
76.1675
4Prime!    1373
19.5915
3222
6163
2Prime!    51
552
111
71
001
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.
9528.736.854.321.862.795.05620
825.806.796.315.964.402.909.204.469.513.697.608.52839
9422.304.559.111.550.001.83920
497.493.357.160.628.207.535.702.826.061.753.394.79439
9322.183.289.860.875.017.85420
492.098.349.051.600.369.131.963.006.150.943.890.29439
9221.109.211.394.290.929.32920
445.598.805.688.862.000.797.000.268.886.508.895.54439
9121.000.000.000.000.000.00920
441.000.000.000.000.000.525.000.000.000.000.000.14439
9020.990.098.888.890.099.00420
440.584.251.365.385.368.919.863.583.563.152.485.04439
899.000.000.000.000.000.00219
81.000.000.000.000.000.099.000.000.000.000.000.01838
882.940.416.984.997.499.91619
8.646.052.045.661.787.666.667.871.665.402.506.46837
872.110.423.522.968.524.90419
4.453.887.446.298.879.977.799.788.926.447.883.54437
862.100.000.000.000.000.00919
4.410.000.000.000.000.052.500.000.000.000.000.14437
852.099.009.888.890.099.00419
4.405.842.513.658.425.780.875.248.563.152.485.04437
842.098.890.099.009.889.00419
4.405.339.647.721.741.680.861.471.277.469.335.04437
83900.000.000.000.000.00218
810.000.000.000.000.009.900.000.000.000.000.01836
82210.000.000.000.000.00918
44.100.000.000.000.005.250.000.000.000.000.14435
8190.000.000.000.000.00217
8.100.000.000.000.000.990.000.000.000.000.01834
8066.891.629.092.892.82417
4.474.490.042.701.146.116.411.072.400.944.74434
7921.000.000.000.000.00917
441.000.000.000.000.525.000.000.000.000.14433
789.000.000.000.000.00216
81.000.000.000.000.099.000.000.000.000.01832
772.219.536.203.158.25416
4.926.340.957.130.173.710.317.590.436.29431
762.100.000.000.000.00916
4.410.000.000.000.052.500.000.000.000.14431
752.099.198.088.021.02916
4.406.632.612.751.158.511.572.162.366.04431
742.099.009.780.099.00416
4.405.842.056.951.283.821.596.502.485.04431
73945.318.054.069.68615
893.626.223.350.104.401.053.322.626.39830
72900.000.000.000.00215
810.000.000.000.009.900.000.000.000.01830
71895.895.199.033.81715
802.628.207.651.848.848.156.702.826.20830
70667.990.192.880.10915
446.210.897.784.009.900.487.798.012.64430
69299.692.501.995.80715
89.815.595.752.508.880.525.759.551.89829
68295.580.359.679.16215
87.367.749.028.064.846.082.094.776.37829
67294.572.920.786.61615
86.773.205.660.760.006.706.650.237.76829
66284.366.947.570.41715
80.864.560.870.518.281.507.806.546.80829
65222.266.487.782.09915
49.402.391.590.991.519.909.519.320.49429
64210.000.000.000.00915
44.100.000.000.005.250.000.000.000.14429
6390.000.000.000.00214
8.100.000.000.000.990.000.000.000.01828
6222.231.909.422.37414
494.257.796.564.797.465.697.752.49427
6122.199.010.990.21414
492.796.088.943.797.349.880.697.29427
6121.000.000.000.00914
441.000.000.000.525.000.000.000.14427
609.000.000.000.00213
81.000.000.000.099.000.000.000.01826
596.673.701.200.52913
44.538.287.713.988.931.778.283.54426
582.948.655.661.41613
8.694.570.209.621.269.020.754.96825
572.221.874.942.95913
4.936.728.262.164.612.628.276.39425
562.220.153.281.41413
4.929.080.592.988.892.950.809.29425
552.219.890.109.77913
4.927.912.099.510.159.902.197.29425
542.100.000.000.00913
4.410.000.000.052.500.000.000.14425
53900.000.000.00212
810.000.000.009.900.000.000.01824
52291.222.254.25712
84.810.401.376.567.310.401.84823
51284.038.530.77612
80.677.886.967.376.968.877.60823
50210.000.000.00912
44.100.000.005.250.000.000.14423
49209.902.199.00412
44.058.933.148.184.133.985.04423
4892.917.907.01611
8.633.737.444.884.447.373.36822
4790.000.000.00211
8.100.000.000.990.000.000.01822
46Prime!    29.097.613.05711
846.671.085.818.580.176.64821
4522.199.009.77911
492.796.035.323.530.697.29421
4421.000.000.00911
441.000.000.525.000.000.14421
439.000.000.00210
81.000.000.099.000.000.01820
422.984.477.00710
8.907.103.026.203.017.09819
412.219.890.21410
4.927.912.577.752.197.29419
402.100.000.00910
4.410.000.052.500.000.14419
39900.000.0029
810.000.009.900.000.01818
38703.210.9749
494.505.678.876.505.49418
37665.012.0449
442.241.023.320.142.24418
36221.990.2149
49.279.656.665.697.29417
35210.000.0099
44.100.005.250.000.14417
3490.000.0028
8.100.000.990.000.01816
3329.857.8868
891.493.565.394.19815
3229.802.8978
888.212.878.212.88815
3128.867.7228
833.345.575.543.33815
3021.155.3448
447.548.727.845.74415
2921.152.8248
447.442.111.244.74415
2821.117.3048
445.940.676.049.54415
2721.037.9248
442.594.393.495.24415
2621.000.0098
441.000.525.000.14415
259.000.0027
81.000.099.000.01814
248.972.8767
80.512.566.521.50814
232.219.7797
4.927.434.347.29413
222.109.6647
4.450.696.960.54413
212.100.0097
4.410.052.500.14413
20942.9976
889.249.942.98812
19900.0026
810.009.900.01812
18669.0246
447.597.795.74412
17292.9676
85.831.713.85811
16223.1146
49.781.418.79411
15210.0096
44.105.250.14411
1490.0025
8.100.990.01810
1329.6965
882.060.2889
1221.0095
441.525.1449
119.0024
81.099.0188
102.9824
8.913.1987
92.9724
8.853.5887
82.1094
4.462.6447
79023
819.9186
66643
445.5446
52193
49.4945
4262
8583
3Prime!    192
4943
241
442
111
81
001
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.
10723.683.958.404.363.815.50520
560.929.885.699.635.409.979.904.536.996.588.929.06539
10622.133.746.889.973.981.09620
489.902.751.389.432.880.606.088.234.983.157.209.98439
10519.890.991.099.890.991.09920
395.651.526.935.942.619.999.916.249.539.625.156.59339
1046.999.999.999.999.999.99619
48.999.999.999.999.999.999.999.999.999.999.999.98438
1032.412.063.028.201.142.84719
5.818.048.052.014.867.252.527.684.102.508.408.18537
1021.989.000.119.790.001.09919
3.956.121.476.524.638.737.378.364.256.741.216.59337
101699.999.999.999.999.99618
489.999.999.999.999.999.999.999.999.999.999.98436
100294.011.283.367.553.97418
86.442.634.747.436.122.422.163.474.743.624.46835
99238.427.502.869.428.60518
56.847.674.124.551.387.578.315.542.147.674.86535
98Prime!    183.952.041.061.237.03918
33.838.353.410.595.039.093.059.501.435.383.83335
9774.972.390.516.353.90517
5.620.859.339.736.673.553.766.379.339.580.26534
9669.999.999.999.999.99617
4.899.999.999.999.999.999.999.999.999.999.98434
9529.440.591.428.026.31817
866.748.423.631.976.949.679.136.324.847.66833
9424.197.921.094.983.44517
585.539.385.319.044.999.440.913.583.935.58533
9323.672.512.354.237.73717
560.387.841.161.538.474.835.161.148.783.06533
929.108.243.842.099.28416
82.960.105.887.139.599.593.178.850.106.92832
916.999.999.999.999.99616
48.999.999.999.999.999.999.999.999.999.98432
903.004.385.430.296.78116
9.026.331.813.779.597.959.773.181.336.20931
892.913.451.145.839.30416
8.488.197.579.192.376.732.919.757.918.84831
881.835.437.923.830.08916
3.368.832.372.233.722.273.322.732.388.63331
87951.856.579.421.42115
906.030.947.787.855.558.787.749.030.60930
86930.589.355.492.66815
865.996.548.556.266.662.655.845.699.56830
85699.999.999.999.99615
489.999.999.999.999.999.999.999.999.98430
84697.090.484.234.13615
485.935.143.209.787.787.902.341.539.58430
83630.600.416.802.79915
397.656.885.671.868.868.176.588.656.79330
82630.477.098.800.54915
397.501.372.111.962.269.111.273.105.79330
81625.328.055.612.04315
391.035.177.135.543.345.531.771.530.19330
80294.411.411.576.17415
86.678.079.266.277.677.266.297.087.66829
79225.762.237.924.33515
50.968.588.072.605.850.627.088.586.90529
78220.050.252.330.34615
48.422.113.550.650.705.605.531.122.48429
77183.014.335.582.88915
33.494.247.028.847.774.882.074.249.43329
7676.761.240.619.74514
5.892.288.061.483.003.841.608.822.98528
7571.108.945.208.78514
5.056.482.088.706.556.078.802.846.50528
7469.999.999.999.99614
4.899.999.999.999.999.999.999.999.98428
7369.696.500.774.73614
4.857.602.220.243.333.420.222.067.58428
7263.220.385.270.74914
3.996.817.113.782.442.873.117.186.99328
7158.198.234.728.95314
3.387.034.525.566.776.655.254.307.83328
7055.827.247.193.61914
3.116.681.529.217.887.129.251.866.11328
6922.388.140.098.98514
501.228.817.091.959.190.718.822.10527
6818.956.175.005.05914
359.336.570.822.575.228.075.633.95327
676.999.999.999.99613
48.999.999.999.999.999.999.999.98426
666.933.240.247.18613
48.069.820.325.255.252.302.896.08426
653.010.857.767.52113
9.065.264.496.265.626.944.625.60925
641.830.169.111.63913
3.349.518.977.212.127.798.159.43325
63906.998.967.65812
822.647.127.339.933.721.746.22824
62722.945.524.91712
522.650.232.003.300.232.056.22524
61714.048.799.58512
509.865.688.194.491.886.568.90524
60699.999.999.99612
489.999.999.999.999.999.999.98424
59228.640.304.57512
52.276.388.877.977.888.367.22523
58197.920.018.79312
39.172.333.840.604.833.327.19323
5793.910.205.19411
8.819.126.640.330.466.219.18822
5693.236.732.76811
8.693.088.337.997.338.803.96822
5569.999.999.99611
4.899.999.999.999.999.999.98422
5462.884.810.89311
3.954.499.441.551.449.944.59322
53Prime!    18.963.885.55911
359.628.955.646.559.826.95321
527.228.936.82510
52.257.527.677.672.575.22520
516.999.999.99610
48.999.999.999.999.999.98420
501.771.260.81910
3.137.364.903.094.637.31319
49699.999.9969
489.999.999.999.999.98418
48695.334.3769
483.489.900.009.984.38418
47287.876.0589
82.872.627.072.627.82817
46225.207.1859
50.718.277.977.281.70517
45182.901.1039
33.452.814.941.825.43317
4491.842.9388
8.435.125.995.215.34816
4391.050.7848
8.290.245.995.420.92816
4269.999.9968
4.899.999.999.999.98416
4169.692.7368
4.857.078.008.707.58416
4059.318.0098
3.518.626.666.268.15316
3957.524.7538
3.309.097.667.909.03316
3823.352.3278
545.331.363.133.54515
37Prime!    19.782.1998
391.335.555.533.19315
36Prime!    17.869.1698
319.307.343.703.91315
3517.646.6198
311.403.303.304.11315
347.243.2257
52.464.366.346.42514
336.999.9967
48.999.999.999.98414
322.281.8177
5.206.707.076.02513
31Prime Curios!    1.934.0637
3.740.615.160.47313
30896.4786
803.679.976.30812
29699.9966
489.999.999.98412
28597.8836
357.468.864.75312
27298.1446
88.892.229.88811
26283.7786
80.532.223.50811
25241.0976
58.129.692.18511
24228.1756
52.065.656.02511
23188.5836
35.565.056.55311
2293.8445
8.807.447.08810
2169.9965
4.899.999.98410
2057.4895
3.305.445.03310
1930.0315
902.101.2099
1828.3145
801.909.1089
179.3944
88.322.3888
166.9964
48.999.9848
152.8284
8.020.2087
142.2354
5.013.1057
139243
861.1686
127073
505.5056
116963
489.9846
102943
88.7885
92843
82.9285
82253
52.4255
72163
48.3845
6Prime!    1733
31.3135
5912
9.0094
4882
8.4484
3662
4.8844
2Prime!    31
332
111
91
001
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.
7525.210.573.274.064.841.32820
635.573.004.806.992.453.404.354.299.608.400.375.53639
7416.681.463.061.790.801.32720
278.271.209.881.890.936.121.639.098.188.902.172.87239
7315.620.041.971.084.230.44220
243.985.711.178.432.931.060.139.234.871.117.589.34239
7215.445.446.545.435.545.54420
238.561.818.987.906.827.999.728.609.789.818.165.83239
7115.444.456.545.434.555.54420
238.531.237.983.816.285.595.582.618.389.732.135.83239
7015.219.869.175.229.085.44420
231.644.417.711.088.481.737.184.880.117.714.446.13239
691.696.782.851.936.705.26919
2.879.072.046.626.459.090.909.546.266.402.709.78237
681.684.230.962.345.308.79719
2.836.633.934.522.604.993.994.062.254.393.366.38237
671.668.295.293.997.506.03919
2.783.209.187.974.225.124.215.224.797.819.023.87237
661.599.655.235.294.309.48719
2.558.896.871.804.492.662.662.944.081.786.988.55237
651.438.728.178.155.345.52919
2.069.938.770.618.199.676.769.918.160.778.399.60237
64262.139.313.895.938.91318
68.717.019.889.833.594.449.533.898.891.071.78635
63251.096.226.342.076.22818
63.049.314.883.231.178.087.113.238.841.394.03635
62172.610.504.583.658.59218
29.794.386.292.625.225.352.252.629.268.349.79235
6150.853.897.306.350.40417
2.586.118.871.244.833.333.384.421.788.116.85234
6048.313.653.281.364.57217
2.334.209.093.391.910.110.191.933.909.024.33234
5945.108.988.816.777.75417
2.034.820.872.072.180.880.812.702.780.284.30234
5825.240.619.955.431.64817
637.088.895.734.534.555.435.437.598.880.73633
5716.517.616.586.567.12717
272.831.657.700.837.616.738.007.756.138.27233
561.545.039.109.604.07216
2.387.145.850.206.157.516.020.585.417.83231
551.510.415.941.428.60216
2.281.356.316.121.663.661.216.136.531.82231
54458.433.179.591.05915
210.160.980.149.972.279.941.089.061.01230
53165.822.864.864.83915
27.497.222.511.984.148.911.522.279.47229
52153.465.444.564.34715
23.551.642.675.334.043.357.624.615.53229
51147.647.028.000.83215
21.799.644.877.479.797.477.844.699.71229
5052.409.339.938.52714
2.746.738.912.792.552.972.198.376.47228
4950.840.261.816.21214
2.584.732.221.541.441.451.222.374.85228
4816.828.270.461.92214
283.190.686.739.747.937.686.091.38227
474.855.892.951.01913
23.579.696.351.799.715.369.697.53226
464.515.673.445.36213
20.391.306.665.188.156.660.319.30226
452.514.764.874.26813
6.324.042.372.874.782.732.404.23625
44544.332.666.72412
296.298.052.067.760.250.892.69224
43262.573.714.11812
68.944.955.348.084.355.944.98623
42152.730.610.57212
23.326.639.407.070.493.662.33223
4147.972.497.14411
2.301.360.482.662.840.631.03222
4016.700.003.32711
278.890.111.272.111.098.87221
3915.114.678.51411
228.453.506.717.605.354.82221
381.665.597.36410
2.774.214.593.954.124.77219
371.632.565.35710
2.665.269.659.569.625.66219
36Prime!    1.534.664.34710
2.355.194.671.764.915.53219
351.458.904.65910
2.128.402.817.182.048.21219
341.417.320.72910
2.008.798.061.608.978.00219
33514.668.9579
264.884.139.931.488.46218
32497.665.8749
247.671.326.623.176.74218
31164.399.7279
27.027.271.717.272.07217
30151.698.4729
23.012.427.772.421.03217
2954.072.5248
2.923.838.338.383.29216
2852.030.3528
2.707.157.997.517.07216
2749.407.0178
2.441.053.773.501.44216
2648.675.3198
2.369.287.117.829.63216
2526.216.7538
687.318.373.813.78615
2426.211.4388
687.039.717.930.78615
2326.132.5788
682.911.868.119.28615
2225.234.0838
636.759.171.957.63615
2115.336.6448
235.212.787.212.53215
2015.019.6148
225.588.939.885.52215
1914.790.5328
218.759.969.957.81215
1814.674.1848
215.331.808.133.51215
171.547.1477
2.393.677.763.93213
16487.6196
237.776.677.73212
15160.1876
25.661.316.65211
1416.9975
289.050.9829
1315.9045
253.080.3529
125.3924
29.122.1928
111.6944
2.884.8827
101.6644
2.783.8727
91.6394
2.701.0727
82583
68.8865
72483
63.7365
6Prime!    1673
29.3925
5Prime!    1573
26.0625
4Prime!    1373
20.0025
3442
2.3324
2122
2523
1Prime!    21
222
001
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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Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com