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
Octagonal numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only
Normal and Palindromic Octagonals
So far this compilation counts 46 Palindromic Octagonals.
Here is the largest Sporadic Palindromic Octagonal that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ June 16, 2023 ]
This basenumber 1.505.189.034.316.273.989.876.364.558
has 28 digits
yielding the following palindromic octagonal number
6.796.782.087.077.872.316.108.073.397.933.708.016.132.787.707.802.876.976
with a length of 55 digits.
|
A palindromic octagonal numbers can only end with digit 0, 1, 3, 5, 6 and 8.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can only be followed by an even number : 10, 12, 14, 16 or 18
3 can only be followed by 3 : 33
5 can only be followed by an even number : 50, 52, 54, 56 or 58
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
8 can only be followed by 0 : 80
There exist no palindromic octagonals of length
2, 3, 5, 6, 7, 8, 10, 11, 15, 17, 18, 20, 22, 23, 26, 31, 34, 38, 39, 40, 44, 45, 46, 47, 50, 52.
(Sloane's A059870)
Case Octa | Change of variables | CUDApalin parameters | Base Correction |
base | n = m + 1 | A B C → 3 4 1 | base = CUDAbase + 1 |
Sloane's A046183 gives the first numbers that are both Octagonal and Triangular.
1, 21, 11781, 203841, 113123361, ...
Consult also Eric Weisstein's page Octagonal Triangular Number.
Sloane's A036428 gives the first numbers that are both Octagonal and Square.
1, 225, 43681, 8473921, 1643897025, ...
Consult also Eric Weisstein's page Octagonal Square Number.
Sloane's A046189 gives the first numbers that are both Octagonal and Pentagonal.
1, 176, 1575425, 234631320, 2098015778145, ...
Consult also Eric Weisstein's page Octagonal Pentagonal Number.
Sloane's A046192 gives the first numbers that are both Octagonal and Hexagonal.
1, 11781, 113123361, 1086210502741, ...
Consult also Eric Weisstein's page Octagonal Hexagonal Number.
Sloane's A048906 gives the first numbers that are both Octagonal and Heptagonal.
1, 297045, 69010153345, ...
Consult also Eric Weisstein's page Octagonal Heptagonal Number.
Sloane's A048924 gives the first numbers that are both Octagonal and Nonagonal.
1, 631125, 286703855361, 130242107189808901, ...
Consult also Eric Weisstein's page Octagonal Nonagonal Number.
The best way to get a 'structural' insight as how to imagine octagonals is to visit for instance this site :
Sources Revealed
Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
One can find the regular octagonal numbers at
%N Octagonal numbers: n(3n2) under A000567.
The palindromic octagonal numbers are categorised as follows :
%N n(3n2) is a palindromic octagonal number under A057106.
%N Palindromic octagonal numbers under A057107.
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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The Table
Index Nr | Info | Basenumber | Length |
Palindromic Octagonals | Length |
| | |
[PG8] Formula = n(3n2)
|
46 | Info | 1.505.189.034.316.273.989.876.364.558 | 28 |
6.796.782.087.077.872.316.108.073.397.933.708.016.132.787.707.802.876.976 | 55 |
45 | Info | Prime! 1.050.527.056.977.712.014.268.976.327 | 28 |
3.310.821.292.326.758.954.662.668.504.058.662.664.598.576.232.921.280.133 | 55 |
44 | Info | 233.009.668.152.040.466.112.406.343 | 27 |
162.880.516.356.972.063.257.851.225.522.158.752.360.279.653.615.088.261 | 54 |
43 | Info | 134.655.235.447.046.729.292.188.665 | 27 |
54.396.097.299.898.769.932.478.624.342.687.423.996.789.899.279.069.345 | 53 |
42 | Info | 10.504.193.933.024.178.843.891.507 | 26 |
331.014.270.547.745.901.058.874.464.478.850.109.547.745.072.410.133 | 51 |
41 | Info | 1.505.231.393.963.863.016.048.058 | 25 |
6.797.164.648.123.182.571.630.290.920.361.752.813.218.464.617.976 | 49 |
40 | Info | 1.498.828.240.862.001.472.469.376 | 25 |
6.739.458.286.816.445.701.381.236.321.831.075.446.186.828.549.376 | 49 |
39 | Info | 1.057.171.651.503.698.963.468.087 | 25 |
3.352.835.702.229.174.992.660.413.140.662.994.719.222.075.382.533 | 49 |
38 | Info | 516.618.664.116.856.556.011.482 | 24 |
800.684.532.341.656.355.118.723.327.811.553.656.143.235.486.008 | 48 |
37 | Info | 1.061.445.921.245.492.013.317 | 22 |
3.380.002.331.186.073.700.219.120.073.706.811.332.000.833 | 43 |
36 | Info | 790.350.914.396.051.463.771 | 21 |
1.873.963.703.660.024.006.763.676.004.200.663.073.693.781 | 43 |
35 | Info | 736.230.589.620.506.244.343 | 21 |
1.626.106.443.278.874.830.658.560.384.788.723.446.016.261 | 43 |
34 | Info | 332.369.949.023.486.562.727 | 21 |
331.409.349.041.625.168.935.539.861.526.140.943.904.133 | 42 |
33 | Info | 246.919.023.530.449.916.313 | 21 |
182.907.012.543.692.638.362.263.836.296.345.210.709.281 | 42 |
32 | Info | 65.621.182.548.683.007.153 | 20 |
12.918.418.797.262.737.739.193.773.726.279.781.481.921 | 41 |
31 | Info | 704.687.633.763.271.761 | 18 |
1.489.753.983.536.637.090.907.366.353.893.579.841 | 37 |
30 | Info | 589.588.144.333.920.233 | 18 |
1.042.842.539.817.346.669.666.437.189.352.482.401 | 37 |
29 | Info | 221.420.450.951.946.111 | 18 |
147.081.048.299.289.519.915.982.992.840.180.741 | 36 |
28 | Info | 145.815.635.485.061.318 | 18 |
63.786.598.655.736.820.002.863.755.689.568.736 | 35 |
27 | Info | 136.626.014.502.187.505 | 18 |
56.000.003.516.255.851.015.855.261.530.000.065 | 35 |
26 | Info | 7.897.699.388.697.563 | 16 |
187.120.966.902.701.565.107.209.669.021.781 | 33 |
25 | Info | 4.731.413.168.091.258 | 16 |
67.158.811.701.562.055.026.510.711.885.176 | 32 |
24 | Info | 222.032.718.762.723 | 15 |
147.895.584.603.498.894.306.485.598.741 | 30 |
23 | Info | 105.802.560.494.387 | 15 |
33.582.545.421.505.050.512.454.528.533 | 29 |
22 | Info | 78.849.864.240.621 | 14 |
18.651.903.272.292.929.227.230.915.681 | 29 |
21 | Info | 51.722.791.547.842 | 14 |
8.025.741.496.504.444.056.941.475.208 | 28 |
20 | Info | 5.847.307.263.801 | 13 |
102.573.006.711.888.117.600.375.201 | 27 |
19 | Info | 1.400.295.262.095 | 13 |
5.882.480.463.134.313.640.842.885 | 25 |
18 | Info | 335.038.979.077 | 12 |
336.753.352.502.205.253.357.633 | 24 |
17 | Info | 13.175.129.925 | 11 |
520.752.145.595.541.257.025 | 21 |
16 | Info | 1.050.553.507 | 10 |
3.310.988.011.108.890.133 | 19 |
15 | Info | 42.687.015 | 8 |
5.466.543.663.456.645 | 16 |
14 | Info | 2.017.631 | 7 |
12.212.500.521.221 | 14 |
13 | Info | 1.472.746 | 7 |
6.506.939.396.056 | 13 |
12 | Info | 1.054.167 | 7 |
3.333.802.083.333 | 13 |
11 | Info | 1.054.067 | 7 |
3.333.169.613.333 | 13 |
10 | Info | 453.486 | 6 |
616.947.749.616 | 12 |
9 | Info | 426.115 | 6 |
544.721.127.445 | 12 |
8 | Info | 7.863 | 4 |
185.464.581 | 9 |
7 | Info | 7.341 | 4 |
161.656.161 | 9 |
6 | Info | 6.431 | 4 |
124.060.421 | 9 |
5 | Info | 6.331 | 4 |
120.232.021 | 9 |
4 | Info | 52 | 2 |
8.008 | 4 |
3 | Info | Prime! 2 | 1 |
8 | 1 |
2 | Info | 1 | 1 |
1 | 1 |
1 | Info | 0 | 1 |
0 | 1 |
Contributions
The Lowell Family (email) submitted the first 8 palindromic octagonals.
Feng Yuan (email) from Washington State, USA, discovered the palindromic octagonals
starting from index number [18] up to [31].
[ January 3, 2008 ]
Feng Yuan (email) from Washington State, USA, discovered the palindromic octagonals
with index numbers [32] and [36].
Exhaustive search performed up to length 43 by Patrick De Geest and found [37] on [ July 17, 2022 ].
On [ July 5, 2022 ] PDG discovered [35] 736230589620506244343 which is smaller than Feng Yuan's record.
On [ July 30, 2022 ] PDG discovered [33] 246919023530449916313 which is smaller than Feng Yuan's record.
On [ July 31, 2022 ] PDG discovered [34] 332369949023486562727 which is smaller than Feng Yuan's record.
So it seems Feng Yuan didn't search exhaustively !
[ November 9, 2022 ]
David Griffeath (email) took over and confirmed voids for 44, 45, 46 and 47-digit palindromic octagonals,
and shared his 48-digit palindromic octagonal record [38].
[ December 6, 2022 ]
Robert Xiao (email) added four new entries [39] up to [42].
[ January 8, 2023 ]
David Griffeath's (email) laptop cranked out an exhaustive search of 53-digit sporadic octagonals.
[43] is his unique find, a new record.
[ June 15 & 16, 2023 ]
Patrick De Geest (email) using CUDApalin from R. Xiao added three more entries [44] up to [46].
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