A195207 Number of even divisors of !n.

0, 0, 0, 1, 0, 4, 0, 6, 0, 24, 0, 4, 0, 24, 0, 4, 0, 16, 0, 24, 0, 32, 0, 96, 0, 144, 0, 16, 0, 64, 0, 144, 0, 40, 0, 128, 0, 160, 0, 16, 0, 96, 0, 96, 0, 32, 0, 128, 0, 96, 0, 48, 0, 128, 0, 96, 0, 96, 0, 32, 0

Offset: 0

Author

Michel Lagneau, Sep 13 2011

Keywords

• nonn

Comments

!n is a subfactorial number (A000166).

Property of this sequence : for n different of 3, the number of even divisors of !n seems even.

From Robert Israel, Jul 31 2024: (Start)

a(n) = 0 if n is even, a(n) = A000005(A000166(n)/2) if n is odd.

Since n - 1 | A000166(n), a(n) >= A000005((n-1)/2) for odd n. (End)

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 0..82

Crossrefs

Cf. A000005, A000166, A183063, A195208, A195209, A195210.

Programs

Maple

A166 := proc(n) option remember; (n-1)*(procname(n-1)+procname(n-2)); end:
A166(0):= 1: A166(1):= 0:
f:= proc(n) if n::even then 0 else numtheory:-tau(A166(n)/2) fi end proc:
map(f, [$0...60]); # Robert Israel, Jul 31 2024

Mathematica

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

Formula

a(n) = A183063(A000166(n)), for n != 1. - Amiram Eldar, Aug 02 2024