cp's OEIS Frontend

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.

%I A171775 #61 Aug 05 2015 21:50:50
%S A171775 1,3,5,52,130,1885,1073741824,4398046511104
%N 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.
%C A171775 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
%e A171775 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).
%Y A171775 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.
%K A171775 nonn,base,hard,more
%O A171775 1,2
%A A171775 _James G. Merickel_, Dec 18 2009
%E A171775 a(7) and a(8) added by _James G. Merickel_, Feb 04 2010
%E A171775 Offset changed to 1, with corresponding addition of a(1) by _James G. Merickel_, Jul 24 2015
%E A171775 Comment corrected and explained._James G. Merickel_, Aug 05 2015
%E A171775 Definition and example rewritten by _N. J. A. Sloane_, Aug 05 2015