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A369308 The number of square divisors d of n such that n/d is also a square.

%I A369308 #14 Jan 19 2024 20:01:46
%S A369308 1,0,0,2,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,
%T A369308 0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,
%U A369308 0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0
%N A369308 The number of square divisors d of n such that n/d is also a square.
%H A369308 Amiram Eldar, <a href="/A369308/b369308.txt">Table of n, a(n) for n = 1..10000</a>
%F A369308 Multiplicative with a(p^e) = 0 is e is odd, and e/2 + 1 if e is even.
%F A369308 a(n) = 0, if n is not a square (A000037), and A000005(sqrt(n)) otherwise.
%F A369308 Dirichlet g.f.: zeta(2*s)^2.
%F A369308 Sum_{k=1..n} a(k) ~ sqrt(n) * (log(n)/2 + 2*gamma - 1), where gamma is Euler's constant (A001620).
%t A369308 f[p_,e_] := If[OddQ[e], 0, e/2+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A369308 (PARI) a(n) = vecprod(apply(x -> if(x%2, 0, x/2 + 1), factor(n)[,2]));
%o A369308 (Python)
%o A369308 from math import prod
%o A369308 from sympy import factorint
%o A369308 def A369308(n): return prod(0 if e&1 else (e>>1)+1 for e in factorint(n).values()) # _Chai Wah Wu_, Jan 19 2024
%Y A369308 Cf. A000005, A000037, A001620, A046951, A369307.
%K A369308 nonn,easy,mult
%O A369308 1,4
%A A369308 _Amiram Eldar_, Jan 19 2024