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

1,3

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

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A052064, A052061, A052062, A050742.

Sequence in context: A101271 A093110 A165707 * A129525 A351987 A295165

Adjacent sequences: A052060 A052061 A052062 * A052064 A052065 A052066

Keyword

nonn,base

Author

Patrick De Geest, Jan 15 2000

Status

approved