A114255 Numbers that are nontrivial (3 digits or more) palindromes when expressed in some base 2 or greater.

5, 7, 9, 10, 13, 15, 16, 17, 20, 21, 23, 25, 26, 27, 28, 29, 31, 33, 34, 36, 37, 38, 40, 41, 42, 43, 45, 46, 49, 50, 51, 52, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 78, 80, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 97, 98, 99, 100, 101, 104, 105, 107, 109

Offset: 1

Author

Jason Orendorff (jason.orendorff(AT)gmail.com), Feb 05 2006

Keywords

• easy
• base
• nonn

Comments

All integers are trivially palindromes in base 1. All integers n>2 are trivially 2-digit palindromes because they can be represented as "11" in base n-1.

Examples

			5 is present because the palindrome (101 base 2) = 5; 803 is present because (30203 base 4) = 803.
		

Crossrefs

Cf. A060873-A060879, A060947-A060949, A123586.

Programs

Haskell

isPalindrome s = (s == reverse s) digits 0 _ = [] digits n b = n `rem` b : digits (n `quot` b) b check n = any isPalindrome $ takeWhile (\x -> length x > 2) $ map (digits n) [2..] main = mapM print $ filter check [1..]

Mathematica

palindromeQ[n_, b_] := (id = IntegerDigits[n, b]) === Reverse[id] && Length[id] >= 3; palindromeQ[n_] := Or @@ (palindromeQ[n, #] & ) /@ Range[2, n-2]; Select[ Range[110], palindromeQ] (* Jean-François Alcover, Dec 16 2011 *)

PARI

isok(n) = for (b=2, n-1, if ((d=digits(n,b)) && (#d >= 3) && (Vecrev(d) == d), return (1));); \\ Michel Marcus, Jul 28 2016

Extensions

Cross-references from Charles R Greathouse IV, Aug 04 2010