A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome.

0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 2, 2, 2, 0, 2, 3, 1, 1, 3, 1, 3, 2, 3, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 2, 3, 3, 0, 4, 1, 3, 3, 4, 0, 3, 3, 3, 3, 1, 1, 5, 1, 2, 4, 3, 4, 3, 2, 3, 1, 3, 1, 5, 2, 2, 2, 2, 1, 5, 0, 6, 2, 3, 1, 5, 4, 2, 1, 4, 1, 4, 3, 4, 3, 1, 1, 5, 1, 4, 3, 6, 1, 3, 0, 5

Offset: 0

Author

John P. Linderman, Feb 26 2008, Feb 28 2008

Keywords

• nonn
• base

Comments

Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11). So here we assume 1 < b < n-1.

Records for a(n)>=1 are in A107129. - Dmitry Kamenetsky, Oct 22 2015

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• John P. Linderman, Description of A135549-A135551 and A016038
• John P. Linderman, Perl program [Use the command: palin.pl]

Crossrefs

Cf. A016038, A037183, A065531, A107129, A135550, A135551.

Cf. A016038 (non-palindromic numbers in any base 1 < b < n-1)

Programs

Mathematica

a = {0, 0, 0}; For[n = 4, n < 100, n++, c = 0; For[b = 2, b < n - 1, b++, If[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]], c++ ]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Feb 27 2008 *)
Table[cnt=0; Do[d=IntegerDigits[n,b]; If[d==Reverse[d], cnt++ ], {b,2,n-2}]; cnt, {n,0,100}] (* T. D. Noe, Feb 28 2008 *)
Table[Total[Boole[Table[PalindromeQ[IntegerDigits[n,b]],{b,2,n-2}]]],{n,0,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 14 2020 *)

Formula

a(n) = A065531(n)-1 = A126071(n)-2 for n>2. - T. D. Noe, Feb 28 2008