A306238 - OEIS

%I #22 Mar 04 2019 17:26:07

%S 1,1,2,2,7,7,11,13,11,11,19,13,37,17,19,17,19,19,31,41,23,23,31,31,37,

%T 29,31,29,31,31,37,37,53,37,53,37,41,41,61,41,43,43,47,61,47,47,53,53,

%U 53,53,59,53,61,61,59,67,59,59,83,61,73,67,67,89,67,67,83,79,71,71,83

%N The smallest integer k such that floor(n!/k) is an odd number.

%C a(n) is usually smaller than 2*n, but there are exceptions, such as n = 0, 12, 19.

%C Are there any other exceptions?

%C Conjecture: for n > 1, a(n) is a prime.

%C a(n) > n if n > 3. - _Robert Israel_, Mar 04 2019

%H Robert Israel, <a href="/A306238/b306238.txt">Table of n, a(n) for n = 0..10000</a>

%F For any k >= 4, a(A040976(k)) = A000040(k).

%e 4! = 24, for k = 1, 2, 3, 4, 5, 6, floor(24/k) are even numbers, floor(24/7) = 3 is an odd number. So a(4) = 7.

%p f:= proc(n) local k,t;

%p   t:= n!;

%p   for k from 1 while floor(t/k)::even do od:

%p   k

%p end proc:

%p map(f, [$0..100]); # _Robert Israel_, Mar 04 2019

%o (PARI) a(n) = {k=1;m=n!;while(floor(m/k)%2==0,k++);k;}

%Y Cf. A000040, A000142 (k!), A040976.

%K nonn

%O 0,3

%A _Jinyuan Wang_, Mar 01 2019