A106404 Number of even semiprimes dividing n.

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0

Offset: 1

Author

Reinhard Zumkeller, May 02 2005

Keywords

• nonn

Comments

Also the number of prime divisors p|n such that n/p is even. - Gus Wiseman, Jun 06 2018

Examples

			a(60) = #{4, 6, 10} = #{2*2, 2*3, 2*5} = 3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000607, A001221, A001227, A008683, A083399, A088705, A100484, A205745, A305614.

Programs

Haskell

a106404 n = length [d | d <- takeWhile (<= n) a100484_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 31 2012

Mathematica

Table[Length[Select[Divisors[n],PrimeQ[#]&&EvenQ[n/#]&]],{n,100}] (* Gus Wiseman, Jun 06 2018 *)
Table[Count[Divisors[n],?(EvenQ[#]&&PrimeOmega[#]==2&)],{n,110}] (* _Harvey P. Dale, May 04 2021 *)
a[n_] := If[EvenQ[n], PrimeNu[n/2], 0]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

PARI

a(n)=if(n%2,0,omega(n/2)) \\ Charles R Greathouse IV, Jan 30 2012

Sage

def A106404(n):
    return add(1-(n/d)%2 for d in divisors(n) if is_prime(d))
print([A106404(n) for n in (1..105)])  # Peter Luschny, Jan 30 2012

Formula

a(n) = A086971(n) - A106405(n).

a(A100484(n)) = 1.

a(A005408(n)) = 0.

a(A005843(n)) > 0 for n>1.

a(2n) = omega(n), a(2n+1) = 0, where omega(n) is the number of distinct prime divisors of n, A001221. - Franklin T. Adams-Watters, Jun 09 2006

a(n) = card { d | d*p = n, d even, p prime }. - Peter Luschny, Jan 30 2012

O.g.f.: Sum_{p prime} x^(2p)/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018