A308657 Smallest number that is nontrivially palindromic in n consecutive number bases.

3, 10, 178

Offset: 1

Author

Matej Veselovac, Jun 14 2019

Keywords

• nonn
• bref
• hard
• base

Comments

Nontrivially palindromic means having at least 2 digits in the palindromic base representation.

| n | term | consecutive palindromic bases representations |

+---+------+-----------------------------------------------+

| 1 | 3 | 11_2 |

| 2 | 10 | 101_3 = 22_4 |

| 3 | 178 | 454_6 = 343_7 = 262_8 |

It is not known if the fourth term exists. The problem can be looked at in context of Diophantine equations, which seem hard.

Examples

			a(1) = 3 because it is the smallest nontrivial palindrome in some number base: 11 when written in binary.
a(2) = A279092(1) = 10 because it is the smallest nontrivial palindrome in two consecutive number bases, namely, bases 3 and 4: 101 and 22 when written in those number bases, respectively.
a(3) = A279093(1) = 178 since it can be written as a palindrome, in three consecutive number bases, and it is the smallest such number. Those bases are 6, 7, 8 and those representations are 454, 343, 262.
		

Links

• Mathoverflow, Can a number be palindromic in more than 3 consecutive number bases?

Crossrefs

Cf. A002113 (palindromes in base 10).

Cf. A279092, A279093 (numbers that are nontrivially palindromic in k or more consecutive integer bases with k=2,3; for k>=4, no examples are known).

Programs

Mathematica

aQ[n_, m_] := SequenceCount[Length[(d = IntegerDigits[n, #])] > 1 && PalindromeQ[d] & /@ Range[2, Ceiling[Sqrt[n]]], Table[True, {m}]] > 0; a[m_] := Module[{n = 2}, While[!aQ[n, m], n++]; n]; Array[a, 3] (* Amiram Eldar, Jul 19 2019 *)