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

0,6

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)

Links

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

Formula

a(n) = A183063(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 even divisors 2, 6, 18, 206, 618, 1854.

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}]

Crossrefs

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

Sequence in context: A023891 A075091 A132953 * A157721 A085968 A353005

Adjacent sequences: A195204 A195205 A195206 * A195208 A195209 A195210

Keyword

nonn

Author

Michel Lagneau, Sep 13 2011

Status

approved