Gobs and gobs of palindromes, base 7/2

Strictly speaking it’s not base 7/2 that I’m using, because I’m allowing digits 0, 1, 2, 3, 4, 5, 6.

However, by using an exploding dots $2 \leftarrow 7$ machine, we can represent positive integers in the form $a_nb^n+a_{n-1}b^{n-1}+\ldots + a_1b+a_0$ where $b=7/2$ and each $a_k=0,1, 2, 3, 4, 5 \textrm{ or } 6$ .

Here, for example is how we work out the 7/2 representation of the base 10 number 37:

So just as for sesquinary palindromes, in which the digits of a palindrome, base 3/2, read the same backwards as forwards, we can look for palindromes base 7/2.

The following Mathematica code produces the base 7/2 digits of a positive integer:

SevenToTwoDigits[n0_] := Module[{n = n0}, digits = {};
While[n >= 7, digits = Prepend[digits, 7*FractionalPart[n/7]];
n = 2*IntegerPart[n/7]];
digits = Prepend[digits, n]]

For example,

SevenToTwoDigits[373]

gives {2, 1, 2, 1, 2}, so the base 10 number 373 is palindromic base 7/2.

There aren’t many sesquinary palindromes, so how many palindromes base 7/2 are there?

Gobs and gobs!

Here’s how we can find base 7/2 palindromes up to 100000:

L = {};
While[n <= 100000,
If[SevenToTwoDigits[n] == Reverse[SevenToTwoDigits[n]], L = {L, n}];
n++]
L = Flatten[L]

This gives us the following 115 numbers (written base 10) up to 100000 that are base 7/2 palindromes:

{1, 2, 3, 4, 5, 6, 9, 18, 27, 30, 37, 44, 53, 60, 67, 74, 83, 90, 97, 135, 207, 270, 279, 342, 373, 422, 478, 534, 583, 641, 697, 746, 802, 851, 907, 956, 965, 1014, 1070, 1119, 1175, 1206, 2151, 2160, 2412, 3357, 3366, 3618, 4293, 5028, 5224, 5427, 5966, 6162, 6365, 6904, 7100, 7296, 7648, 7844, 8383, 8586, 8782, 9321, 9517, 9720, 10455, 10658, 10854, 11393, 11589, 11941, 12137, 12879, 13075, 13271, 13810, 14013, 14209, 14748, 26856, 27738, 28620, 30195, 31077, 31959, 32841, 34416, 35298, 36180, 37755, 38637, 39519, 40401, 51627, 58879, 69225, 69911, 72508, 73194, 76477, 77163, 78395, 79081, 81678, 82364, 85647, 86333, 88930, 89616, 93398, 94084, 94770, 97367, 98053}

Up to 1000000 there are 197 positive integers that are base 7/2 palindromes.

The relative abundance of base 7/2 palindromes has to have something to do with the distance 5=7-2 between 7 and 2, but I’m not sure what yet. Any clues?