A052063 Numbers k such that the decimal expansion of k^3 contains no palindromic substring except single digits.

0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 16, 17, 18, 19, 21, 22, 24, 25, 27, 28, 29, 32, 33, 35, 37, 38, 39, 41, 43, 44, 47, 51, 57, 59, 65, 66, 69, 73, 75, 76, 84, 88, 93, 94, 97, 102, 108, 109, 115, 116, 123, 125, 128, 133, 134, 135, 139, 144, 145, 147, 148, 155, 156, 159

Offset: 1

Author

Patrick De Geest, Jan 15 2000

Keywords

• nonn
• base

Comments

Leading zeros in substring are allowed so 52^3 = 140608 is rejected because 14{060}8 contains a palindromic substring.

Probabilistic analysis strongly suggests that this sequence is not finite. - Franklin T. Adams-Watters, Nov 15 2006

Examples

			19^3 = 6859 -> substrings 68, 85, 59, 685, 859 and 6859 are all non-palindromic.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A052064, A052061, A052062, A050742.

Programs

Mathematica

testQ@l_ :=
NoneTrue[Flatten[Table[Partition[l, n, 1], {n, 2, Length@l}], 1],
  PalindromeQ];
f@nn_ := Select[Range@nn, testQ@IntegerDigits@(#^3) &]; f[300]
(* Hans Rudolf Widmer, May 13 2022 *)

Python

def nopal(s): return all(ss != ss[::-1] for ss in (s[i:j] for i in range(len(s)-1) for j in range(i+2, len(s)+1)))
def ok(n): return nopal(str(n**3))
print([k for k in range(160) if ok(k)]) # Michael S. Branicky, May 13 2022