cp's OEIS Frontend

A370080 The product of the even exponents of the prime factorization of n.

%I A370080 #8 Feb 11 2024 04:53:54
%S A370080 1,1,1,2,1,1,1,1,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,1,2,1,1,2,1,1,1,1,1,1,
%T A370080 1,4,1,1,1,1,1,1,1,2,2,1,1,4,2,2,1,2,1,1,1,1,1,1,1,2,1,1,2,6,1,1,1,2,
%U A370080 1,1,1,2,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A370080 The product of the even exponents of the prime factorization of n.
%H A370080 Amiram Eldar, <a href="/A370080/b370080.txt">Table of n, a(n) for n = 1..10000</a>
%F A370080 a(n) = A005361(A350388(n)).
%F A370080 Multiplicative with a(p^e) = e if e is even, and 1 if e is odd.
%F A370080 a(n) = A005361(n)/A370079(n).
%F A370080 a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
%F A370080 a(n) <= A005361(n), with equality if and only if n is in A335275.
%F A370080 Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s) + 1/p^(4*s)).
%F A370080 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 +  2/p^4 - 1/p^5) = 1.53318063378623623841... .
%F A370080 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + (p^(2*s) + 1)/(p^s*(p^s - 1)*(p^s + 1)^2)). - _Vaclav Kotesovec_, Feb 11 2024
%t A370080 f[p_, e_] := If[EvenQ[e], e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A370080 (PARI) a(n) = vecprod(apply(x -> if(x%2, 1, x), factor(n)[, 2]));
%Y A370080 Cf. A005361, A013661, A268335, A335275, A350386, A350388, A370079.
%K A370080 nonn,easy,mult
%O A370080 1,4
%A A370080 _Amiram Eldar_, Feb 09 2024