A195208 Number of odd divisors of !n.

1, 0, 1, 1, 3, 2, 4, 6, 8, 8, 10, 4, 6, 12, 8, 4, 24, 4, 8, 24, 32, 16, 24, 96, 32, 48, 24, 16, 384, 32, 64, 144, 16, 8, 36, 128, 16, 80, 24, 16, 96, 32, 16, 96, 4, 16, 448, 128, 64, 24, 192, 48, 192, 64, 16, 96, 768, 32, 96, 32, 64

Offset

0,5

Comments

!k is a subfactorial number (A000166).

Links

Amiram Eldar, Table of n, a(n) for n = 0..81

Formula

a(n) = A000005(A000265(A000166(n))). - Robert Israel, Aug 01 2024

a(n) = A001227(A000166(n)), for n != 1. - Amiram Eldar, Aug 02 2024

Example

a(7) = 6 because the divisors of  !7 = 1854 are  {1, 2, 3, 6, 9, 18, 103, 206, 309, 618, 927, 1854} with 6  odd divisors  1, 3, 9, 103, 309 and 927 => 6 is in the sequence.

Maple

A166 := proc(n) option remember; (n-1)*(procname(n-1)+procname(n-2)); end:
A166(0):= 1: A166(1):= 0:
f:= proc(n) local v; v:= A166(n); numtheory:-tau(v/2^padic:-ordp(v, 2)) end proc:
map(f, [$1..60]); # Robert Israel, Aug 01 2024

Mathematica

f[n_] := Block[{d = Divisors[Subfactorial[n]]}, Count[OddQ[d], True]]; Table[f[n], {n, 0, 60}]

Prog

(PARI) a(n)=if(n<2, 1-n, my(x='x, k=n!*polcoeff(exp(-x+x*O(x^n))/(1-x), n)); numdiv(k>>valuation(k, 2))) \\ Charles R Greathouse IV, Sep 13 2011

Crossrefs

Cf. A000005, A000166, A000265, A001227, A195207, A195209, A195210.

Sequence in context: A105746 A166309 A349472 * A114651 A396926 A340064

Adjacent sequences: A195205 A195206 A195207 * A195209 A195210 A195211

Keyword

nonn

Author

Michel Lagneau, Sep 13 2011

Status

approved