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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

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

base x ( 5 x base - 3 )
-------------------------------
2

 PLAIN TEXT POLYGONS

Normal and Palindromic Heptagonals

So far this compilation counts 84 Palindromic Heptagonals.
Here is the largest Sporadic Palindromic Heptagonal that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ June 16, 2023 ]

 This basenumber1.873.720.564.765.469.611.496.441.763 has 28 digits yielding the following palindromic heptagonal number 8.777.071.887.062.576.002.354.728.033.308.274.532.006.752.607.881.707.778 with a length of 55 digits.

A palindromic heptagonal numbers can end with any digit.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
2 can be followed by any number : 20, 21, 22, 23, 24, 25, 26, 27, 28 or 29
3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39
4 can be followed by any number : 40, 41, 42, 43, 44, 45, 46, 47, 48 or 49
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can be followed by any number : 60, 61, 62, 63, 64, 65, 66, 67, 68 or 69
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79
8 can be followed by any number : 80, 81, 82, 83, 84, 85, 86, 87, 88 or 89
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99

There exist no palindromic heptagonals of length  8, 9, 14, 16, 32, 38, 44, 45, 46, 51, 52.
(Sloane's A059869)

 Case Hepta Change of variables CUDApalin parameters Base Correction odd base n = 2 * m + 1 `A B C →  10 7 1 ` CUDAbase * 2 + 1 even base n = 2 * m + 2 `A B C →  10 17 7 ` CUDAbase * 2 + 2

Sloane's A046194 gives the first numbers that are both Heptagonal and Triangular.
1, 55, 121771, 5720653, 12625478965, ...
Consult also Eric Weisstein's page Heptagonal Triangular Number.
Sloane's A036354 gives the first numbers that are both Heptagonal and Square.
1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, ...
Consult also Eric Weisstein's page Heptagonal Square Number.
Sloane's A048900 gives the first numbers that are both Heptagonal and Pentagonal.
1, 4347, 16701685, 246532939589097, ...
Consult also Eric Weisstein's page Heptagonal Pentagonal Number.
Sloane's A048903 gives the first numbers that are both Heptagonal and Hexagonal.
1, 121771, 12625478965, 1309034909945503, ...
Consult also Eric Weisstein's page Heptagonal Hexagonal Number.
Sloane's A048906 gives the first numbers that are both Heptagonal and Octagonal.
1, 297045, 69010153345, ...
Consult also Eric Weisstein's page Heptagonal Octagonal Number.
Sloane's A048921 gives the first numbers that are both Heptagonal and Nonagonal.
1, 26884, 542041975, 10928650279834, ...
Consult also Eric Weisstein's page Heptagonal Nonagonal Number.

The best way to get a 'structural' insight as how to imagine heptagonals is to visit for instance this site :

Interesting semiprime (two distinct primefactors of equal length)
PG7{76} = 1159363692769172939242993 * 2898409231922932348107481

Sources Revealed

 Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences One can find the regular heptagonal numbers at %N Heptagonal (or 7-gonal) numbers n(5n–3)/2 under A000566. The palindromic heptagonal numbers are categorised as follows : %N n(5n–3)/2 is a palindromic heptagonal number under A054971. %N Palindromic heptagonal numbers under A054910. 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.

The Table

Index NrInfoBasenumberLength
Palindromic HeptagonalsLength

[PG7] Formula = n(5n–3)/2
84Info1.873.720.564.765.469.611.496.441.76328
8.777.071.887.062.576.002.354.728.033.308.274.532.006.752.607.881.707.77855
83Info1.865.550.031.166.736.002.957.872.18828
8.700.692.296.965.524.180.003.874.993.994.783.000.814.255.696.922.960.07855
82Info518.868.986.913.124.343.115.029.75127
673.062.563.950.630.002.104.548.005.500.845.401.200.036.059.365.260.37654
81Info136.288.963.956.567.334.982.488.78427
46.436.704.240.886.275.409.737.763.336.773.790.457.268.804.240.763.46453
80Info81.051.618.456.301.707.752.402.68626
16.423.412.135.964.769.068.005.026.562.050.086.096.746.953.121.432.46153
79Info5.491.380.065.793.004.252.725.10225
75.388.137.567.471.949.288.446.366.364.488.294.917.476.573.188.35750
78Info2.506.873.575.346.207.242.110.50125
15.711.037.806.922.690.496.401.855.810.469.409.622.960.873.011.75150
77Info1.401.206.868.817.841.389.000.13925
4.908.451.723.055.748.417.680.431.340.867.148.475.503.271.548.09449
76InfoPrime!    1.159.363.692.769.172.939.242.99325
3.360.310.430.278.433.054.615.306.035.164.503.348.720.340.130.63349
75Info885.337.623.337.961.398.168.74124
1.959.556.768.244.275.027.387.891.987.837.205.724.428.676.559.59149
74Info529.794.084.977.256.736.834.47724
701.704.431.192.221.831.007.830.038.700.138.122.291.134.407.10748
73Info184.726.169.766.590.240.131.46824
85.309.394.491.587.795.230.531.913.503.259.778.519.449.390.35847
72InfoPrime!    155.063.328.766.708.895.002.53124
60.111.589.820.531.125.801.081.218.010.852.113.502.898.511.10647
71Info115.095.375.604.446.234.345.99324
33.117.363.713.821.393.460.337.673.306.439.312.831.736.371.13347
70Info1.713.582.486.121.776.417.58222
7.340.912.341.858.220.171.916.191.710.228.581.432.190.43743
69Info1.645.934.853.472.802.341.77622
6.772.753.854.691.338.287.602.067.828.331.964.583.572.77643
68Info1.461.903.709.744.855.845.21022
5.342.906.141.414.429.317.798.977.139.244.141.416.092.43543
67Info1.197.770.348.245.060.425.83822
3.586.634.517.837.733.317.833.387.133.377.387.154.366.85343
66Info839.449.225.728.858.371.62121
1.761.687.506.441.949.539.786.879.359.491.446.057.861.67143
65Info584.961.928.628.774.900.04321
855.451.144.862.739.850.023.320.058.937.268.441.154.55842
64Info149.974.969.674.243.257.99021
56.231.228.821.975.462.213.831.226.457.912.882.213.26541
63Info116.720.282.191.097.780.65321
34.059.060.686.923.744.328.082.344.732.968.606.095.04341
62Info53.133.142.984.397.396.87720
7.057.827.208.501.045.785.995.875.401.058.027.287.50740
61Info26.678.135.232.269.748.22120
1.779.307.248.678.181.132.112.311.818.768.427.039.77140
60InfoPrime Curios!    22.059.819.805.104.914.32120
1.216.589.124.584.247.550.110.557.424.854.219.856.12140
59Info10.317.264.079.491.259.30720
266.114.845.214.901.555.595.555.109.412.548.411.66239
58Info1.743.257.048.371.139.57719
7.597.362.841.739.144.174.714.419.371.482.637.95737
57Info529.189.005.495.906.00218
700.102.508.844.365.081.180.563.448.805.201.00736
56Info386.569.698.677.663.35818
373.590.329.839.348.608.806.843.938.923.095.37336
55Info140.620.708.714.288.53918
49.435.459.298.271.961.416.917.289.295.453.49435
54Info25.901.555.713.068.48617
1.677.226.470.892.976.776.792.980.746.227.76134
53Info8.570.921.185.931.80616
183.651.724.938.636.676.636.839.427.156.38133
52Info1.019.248.876.540.67216
2.597.170.680.823.553.553.280.860.717.95231
51Info627.391.088.906.93415
984.048.946.099.569.965.990.649.840.48930
50Info523.047.977.062.61115
683.947.965.773.223.322.377.569.749.38630
49Info186.276.561.018.50315
86.747.392.962.199.899.126.929.374.76829
48Info139.689.569.097.50415
48.782.939.286.615.651.668.293.928.78429
47Info87.036.116.082.28114
18.938.213.756.720.602.765.731.283.98129
46Info47.892.275.461.25014
5.734.175.122.140.550.412.215.714.37528
45Info43.909.485.680.87914
4.820.107.331.898.228.981.337.010.28428
44Info10.024.819.305.87214
251.242.505.288.444.882.505.242.15227
43Info6.639.848.560.86113
110.218.972.277.909.772.279.812.01127
42Info3.129.887.444.61213
24.490.488.539.844.893.588.409.44226
41Info1.755.516.804.29713
7.704.598.125.420.245.218.954.07725
40Info1.648.402.664.97613
6.793.078.364.747.474.638.703.97625
39Info987.706.147.38712
2.438.908.583.963.693.858.098.34225
38Info498.373.323.65112
620.939.924.316.613.429.939.02624
37Info178.357.794.54212
79.528.757.184.448.175.782.59723
36Info159.052.425.33612
63.244.185.012.921.058.144.23623
35Info147.826.795.94512
54.631.903.998.189.930.913.64523
34InfoPrime Curios!    71.784.986.52111
12.882.710.724.442.701.728.82123
33Info36.987.715.85311
3.420.227.810.000.187.220.24322
32Info29.970.725.86711
2.245.611.022.442.201.165.42222
31Info17.663.032.82211
779.956.821.151.128.659.97721
30Info15.911.875.33611
632.969.441.747.144.969.23621
29Info2.850.685.77210
20.316.023.422.432.061.30220
28Info2.497.949.42610
15.599.378.333.387.399.55120
27Info1.820.198.06310
Only even digits !    8.282.802.468.642.082.82819
26Info765.609.0419
1.465.393.008.003.935.64119
25Info496.329.4169
615.857.222.222.758.51618
24Info88.274.1418
19.480.809.790.808.49117
23Info16.943.2628
717.685.292.586.71715
22Info1.686.5377
7.111.015.101.11713
21Info1.207.6987
3.646.334.336.46313
20Info615.7696
947.927.729.74912
19Info382.1736
365.139.931.56312
18Info371.7786
345.546.645.54312
17Info190.9146
91.120.102.11911
16Info171.6576
73.665.056.63711
15Info161.6566
65.331.413.35611
14Info158.4766
62.786.368.72611
13Info53.5375
7.165.445.61710
12Info1.5564
6.050.5067
11Info1.4704
5.400.0457
10Info1.3594
4.615.1647
9Info5833
848.8486
8Info1563
60.6065
7Info442
4.7744
6Info382
3.5534
5Info162
6163
4InfoPrime!    51
552
3InfoPrime!    21
Prime!    71
2Info11
11
1Info01
01

Contributions

Jeff Heleen (email submitted the first palindromic heptagonals [1] and [53].

Feng Yuan (email from Washington State, USA, discovered the palindromic heptagonals
starting from index number [54] and [55].

[ January 8, 2008 ]
Feng Yuan (email) from Washington State, USA, discovered the palindromic heptagonals
starting from index number [56] up to [71] alas with some missers.

[ November 4, 2022 ]
Robert Xiao (email) added two missing entries [57] and [62].

[ November 11, 2022 ]
Robert Xiao (email) added four missing entries [63], [66], [68] and [70].

[ November 11, 2022 ]
David Griffeath (email) using Rust code by Robert Xiao (email) added entry [74]. This was once a record holder.

[ December 6, 2022 ]
Robert Xiao (email) added two missing entries [72] and [73] and five new entries [75] up to [79]

[ January 11, 2023 ]
David Griffeath (email) using CUDA code by Robert Xiao (email) added two new entries [80] and [81].
An exhaustive search for 53-digit palindromic heptagonals was done. There are two, both with even roots.

[ June 15 & 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [82] up to [84].

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