A195207 - OEIS

%I #23 Aug 02 2024 03:15:23

%S 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,

%T 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,

%U 0,96,0,96,0,32,0

%N Number of even divisors of !n.

%C !n is a subfactorial number (A000166).

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

%C From _Robert Israel_, Jul 31 2024: (Start)

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

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

%H Amiram Eldar, <a href="/A195207/b195207.txt">Table of n, a(n) for n = 0..82</a>

%F a(n) = A183063(A000166(n)), for n != 1. - _Amiram Eldar_, Aug 02 2024

%e 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.

%p A166 := proc(n) option remember; (n-1)*(procname(n-1)+procname(n-2)); end:

%p A166(0):= 1: A166(1):= 0:

%p f:= proc(n) if n::even then 0 else numtheory:-tau(A166(n)/2) fi end proc:

%p map(f, [$0...60]); # _Robert Israel_, Jul 31 2024

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

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

%K nonn

%O 0,6

%A _Michel Lagneau_, Sep 13 2011