A333512 Numbers nontrivially palindromic in exactly three consecutive number bases.

1654123, 4564873, 1687837537501, 195766180573603

Offset: 1

Author

Matej Veselovac, Mar 25 2020

Keywords

• nonn
• hard
• more

Comments

Numbers which are strictly non-palindromic up to a set of k=3 consecutive number bases. It is conjectured that such a sequence for k>=4 is empty. For a special case of k=0, we have the sequence A016038.

A subsequence of A279093. Notice that a(1),a(2),a(3),a(4) are all of the form (b^3 + 3 b^2 + 5 b + 2)/2 where b=2k+6, for k=71,101,7497,36575. Not all terms are necessarily of this form. (See comments in A279093, containing a total of 9 known such forms that generate numbers palindromic in three consecutive number bases.)

For every n, a(n) should be a prime number.

Examples

			N = 1654123 is a palindrome when written in three consecutive number bases b = 148,149,150 and is not a palindrome in any other nontrivial number bases 2 <= b <= N-2. The three palindromic representations are: 1654123 = (75,76,75)_148 = (74,75,74)_149 =  (73,77,73)_150. Hence, this number is a term of the sequence.
		

Crossrefs

Cf. A279092, A279093 (consecutive palindromes), A016038 (strictly non-palindromic numbers), A060792 (palindromes in bases 2 and 3).