A171775 a(n) = smallest number M such that there exist bases b_2, b_3, ..., b_n with the property that M written in base b_k is a k-digit palindrome for all k=2..n.

1, 3, 5, 52, 130, 1885, 1073741824, 4398046511104

Offset: 1

Author

James G. Merickel, Dec 18 2009

Keywords

• nonn
• base
• hard
• more

Comments

a(n) is no more than 2^[(n-1)*(n-2)] for n > 6 (and equals it for n = 7 and 8 at least). The reason for this bound is that for this number for each length from n down to 3 there is at least one power of 2, 2^k, such that in base b = 2^k-1 the binomial expansion of (b+1)^floor([(n-1)*(n-2)]/k) multiplied by the remaining small power of 2 gives a palindromic expression not requiring carries in base b. James G. Merickel, Aug 05 2015

Examples

			a(6)=1885: the bases are 1884 (1885 is 11 in base 1884), 14 (1885 is 989 in base 14), 12 (it is 1111 in base 12), 6 (it is 12421 in base 6), and 4 (it is 131131 in base 4).
		

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165, A087155, A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A253294.

Extensions

a(7) and a(8) added by James G. Merickel, Feb 04 2010

Offset changed to 1, with corresponding addition of a(1) by James G. Merickel, Jul 24 2015

Comment corrected and explained.James G. Merickel, Aug 05 2015

Definition and example rewritten by N. J. A. Sloane, Aug 05 2015