A306238 The smallest integer k such that floor(n!/k) is an odd number.

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, 29, 31, 29, 31, 31, 37, 37, 53, 37, 53, 37, 41, 41, 61, 41, 43, 43, 47, 61, 47, 47, 53, 53, 53, 53, 59, 53, 61, 61, 59, 67, 59, 59, 83, 61, 73, 67, 67, 89, 67, 67, 83, 79, 71, 71, 83

Offset

0,3

Comments

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

Are there any other exceptions?

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

a(n) > n if n > 3. - Robert Israel, Mar 04 2019

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

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

Example

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.

Maple

f:= proc(n) local k, t;
  t:= n!;
  for k from 1 while floor(t/k)::even do od:
  k
end proc:
map(f, [$0..100]); # Robert Israel, Mar 04 2019

Mathematica

Table[Module[{k=1}, While[EvenQ[Floor[n!/k]], k++]; k], {n, 0, 70}] (* Harvey P. Dale, Dec 12 2025 *)

Prog

(PARI) a(n) = {k=1; m=n!; while(floor(m/k)%2==0, k++); k; }
(Python)
from math import factorial
def A306238(n):
    j, f = 1, factorial(n)
    while j <= f:
        k = f//j
        if k&1:
            return j
        j = f//k+1 # Chai Wah Wu, Jun 24 2026

Crossrefs

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

Sequence in context: A357412 A021443 A045923 * A318086 A244049 A271229

Adjacent sequences: A306235 A306236 A306237 * A306239 A306240 A306241

Keyword

nonn

Author

Jinyuan Wang, Mar 01 2019

Status

approved