This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306536 #22 Mar 09 2019 00:41:51
%S A306536 12,3,5,5,3,7,10,3,5,11,3,7,5,3,17,12,3,5,5,3,11,10,3,5,7,3,7,5,3,11,
%T A306536 11,3,5,5,3,13,10,3,5,7,3,10,5,3,17,7,3,5,5,3,11,10,3,5,7,3,10,5,3,7,
%U A306536 7,3,5,5,3,15,7,3,5,12,3,10,5,3,7
%N A306536 The smallest integer k such that floor((2*n)^k/k) is an odd number.
%C A306536 For n > 0, there are infinitely many numbers k such that floor(n^k/k) is an odd number. For odd number n, it's obvious because k can be n^j for any j >= 0. For n = 2, k can be 3*2^j for any j >= 1.
%C A306536 For even number n > 2, k can be A090368(n/2)^j for any j >= 1. Proof: if odd number k makes n^k == 1 (mod k), then n^k = 1 + k*(odd number t), so floor(n^k/k) = t is an odd number. A090368(n/2)^j (j>0) is such k.
%F A306536 a(2^j) <= 12. (This is because floor((2*2^j)^12/12) = floor(2^(12j+10)/3) = (2^(12j+10) - 1)/3 is an odd integer for all j >= 0. - _Jianing Song_, Feb 24 2019)
%F A306536 For n > 1, a(n) <= A090368(n).
%o A306536 (PARI) a(n) = {k=1; while((2*n)^k\k%2==0, k++); k; }
%Y A306536 Cf. A090368.
%K A306536 nonn
%O A306536 1,1
%A A306536 _Jinyuan Wang_, Feb 22 2019