A126071 Number of bases (2 <= b <= n+1) in which n is a palindrome.

1, 1, 2, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 4, 4, 4, 2, 4, 5, 3, 3, 5, 3, 5, 4, 5, 3, 4, 4, 4, 4, 4, 3, 6, 3, 4, 3, 6, 3, 5, 3, 4, 5, 5, 2, 6, 3, 5, 5, 6, 2, 5, 5, 5, 5, 3, 3, 7, 3, 4, 6, 5, 6, 5, 4, 5, 3, 5, 3, 7, 4, 4, 4, 4, 3, 7, 2, 8, 4, 5, 3, 7, 6, 4, 3

Offset: 1

Author

Paul Richards, Mar 01 2007

Keywords

• nonn
• base

Comments

a(n) >= 1, since n will always have a single "digit" in base n+1.

Examples

			From bases 2 to 9 respectively, 8 can be represented as: 1000, 22, 20, 13, 12, 11, 10, 8. Three of those are symmetrical (22, 11, 8) and so a(8) = 3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A016026.

Cf. A016038, A047811 (related to numbers having 2 bases).

Programs

Mathematica

Table[cnt = 0; Do[d = IntegerDigits[n, k]; If[d == Reverse[d], cnt++], {k, 2, n + 1}]; cnt, {n, 100}] (* T. D. Noe, Oct 04 2012 *)
a = Compile[{{n, Integer}}, Module[{b = 2, c = Floor@ If[ IntegerQ@ Sqrt[n + 1], 0, 1], idn, s = Floor@ Sqrt[n + 1] - 1}, c = c + Count[ Mod[n, Range@ s], 0]; s = s + 2; While[ b < s, If[ Reverse[ idn = IntegerDigits[n, b]] == idn, c++]; b++]; c]]; Array[a, 87] (* _Robert G. Wilson v, May 05 2025 *)

PARI

a(n) = sum(k=2, n+1, d = digits(n, k); Vecrev(d) == d); \\ Michel Marcus, Mar 07 2015

Extensions

Extended by T. D. Noe, Oct 04 2012