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A367987 The number of square divisors of the largest unitary divisor of n that is a square.

%I A367987 #8 Dec 07 2023 10:25:23
%S A367987 1,1,1,2,1,1,1,1,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,1,2,1,1,2,1,1,1,1,1,1,
%T A367987 1,4,1,1,1,1,1,1,1,2,2,1,1,3,2,2,1,2,1,1,1,1,1,1,1,2,1,1,2,4,1,1,1,2,
%U A367987 1,1,1,2,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1
%N A367987 The number of square divisors of the largest unitary divisor of n that is a square.
%C A367987 Also, the number of divisors of the square root of the largest unitary divisor of n that is a square.
%H A367987 Amiram Eldar, <a href="/A367987/b367987.txt">Table of n, a(n) for n = 1..10000</a>
%F A367987 Multiplicative with a(p^e) = e/2 + 1 if e is even and 1 otherwise.
%F A367987 a(n) = A046951(A350388(n)).
%F A367987 a(n) = A000005(A071974(n)).
%F A367987 a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
%F A367987 Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
%F A367987 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/((p-1)*(p+1)^2)) = 1.450032... (A335762).
%t A367987 f[p_, e_] := If[EvenQ[e], e/2 + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367987 (PARI) a(n) = vecprod(apply(x -> if(x%2, 1, x/2+1), factor(n)[, 2]));
%Y A367987 Cf. A000005, A046951, A071974, A268335, A335762, A350388, A365401, A367988.
%K A367987 nonn,easy,mult
%O A367987 1,4
%A A367987 _Amiram Eldar_, Dec 07 2023