A375201 a(n) is the sum of the bases b with 1 < b < n-1 in which n is a palindrome.

0, 2, 0, 2, 3, 2, 7, 0, 5, 3, 6, 6, 10, 6, 13, 0, 12, 12, 10, 3, 23, 4, 20, 10, 22, 4, 23, 7, 22, 12, 20, 6, 41, 6, 22, 12, 38, 5, 37, 6, 31, 24, 31, 0, 56, 6, 40, 22, 45, 0, 51, 20, 43, 30, 28, 4, 82, 6, 35, 34, 53, 26, 63, 11, 52, 22, 56, 7, 91, 10, 42, 38, 55, 10, 87, 0, 91, 34, 52, 5, 112, 29

Offset: 4

Author

Robert Israel, Oct 15 2024

Keywords

• nonn
• look

Comments

If n = s*t is composite with s <= t-2, then n = s * (t-1) + s is a two-digit palindrome in base t-1, while s^2 = (s-1)^2 + 2 * (s-1) + 1 is a palindrome in base s-1. Thus a(n) >= sqrt(n)-1 for composite n > 6. On the other hand, there may be infinitely many primes for which a(n) = 0 (see A016038).

Examples

			a(10) = 7 because 10 = 101_3 = 22_4 is a palindrome in bases 3 and 4, and 3 + 4 = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 4..10000

Crossrefs

Cf. A016038, A135549, A375350.

Programs

Maple

ispali:= proc(x, b) local F; F:= convert(x, base, b);
  andmap(t -> F[t] = F[-t], [$1.. nops(F)/2])
end proc:
f:= proc(k) convert(select(b -> ispali(k, b), [$2..k-2]), `+`) end proc:
map(f, [$4 .. 100]);

Python

from sympy.ntheory import is_palindromic
def a(n): return sum(b for b in range(2, n-2) if is_palindromic(n, b))
print([a(n) for n in range(4, 86)]) # Michael S. Branicky, Oct 15 2024