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Number of even semiprimes dividing n.
5

%I #33 Jun 26 2022 02:18:10

%S 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,

%T 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,

%U 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

%N Number of even semiprimes dividing n.

%C Also the number of prime divisors p|n such that n/p is even. - _Gus Wiseman_, Jun 06 2018

%H Reinhard Zumkeller, <a href="/A106404/b106404.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(A100484(n)) = 1.

%F a(A005408(n)) = 0.

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

%F 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

%F a(n) = card { d | d*p = n, d even, p prime }. - _Peter Luschny_, Jan 30 2012

%F O.g.f.: Sum_{p prime} x^(2p)/(1 - x^(2p)). - _Gus Wiseman_, Jun 06 2018

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

%t Table[Length[Select[Divisors[n],PrimeQ[#]&&EvenQ[n/#]&]],{n,100}] (* _Gus Wiseman_, Jun 06 2018 *)

%t Table[Count[Divisors[n],_?(EvenQ[#]&&PrimeOmega[#]==2&)],{n,110}] (* _Harvey P. Dale_, May 04 2021 *)

%t a[n_] := If[EvenQ[n], PrimeNu[n/2], 0]; Array[a, 100] (* _Amiram Eldar_, Jun 26 2022 *)

%o (Sage)

%o def A106404(n):

%o return add(1-(n/d)%2 for d in divisors(n) if is_prime(d))

%o print([A106404(n) for n in (1..105)]) # Peter Luschny, Jan 30 2012

%o (PARI) a(n)=if(n%2,0,omega(n/2)) \\ _Charles R Greathouse IV_, Jan 30 2012

%o (Haskell)

%o a106404 n = length [d | d <- takeWhile (<= n) a100484_list, mod n d == 0]

%o -- _Reinhard Zumkeller_, Jan 31 2012

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

%K nonn

%O 1,12

%A _Reinhard Zumkeller_, May 02 2005