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

1,12

Comments

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

Links

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

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

Example

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

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 *)

Prog

(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
(PARI) a(n)=if(n%2, 0, omega(n/2)) \\ Charles R Greathouse IV, Jan 30 2012
(Haskell)
a106404 n = length [d | d <- takeWhile (<= n) a100484_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 31 2012

Crossrefs

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

Sequence in context: A244415 A104975 A191254 * A292242 A280800 A083889

Adjacent sequences: A106401 A106402 A106403 * A106405 A106406 A106407

Keyword

nonn

Author

Reinhard Zumkeller, May 02 2005

Status

approved