cp's OEIS Frontend

A370079 The product of the odd exponents of the prime factorization of n.

%I A370079 #9 Feb 11 2024 04:44:56
%S A370079 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,3,1,1,1,1,5,1,1,
%T A370079 1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,1,1,1,
%U A370079 1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A370079 The product of the odd exponents of the prime factorization of n.
%C A370079 First differs from A363329 at n = 32.
%H A370079 Amiram Eldar, <a href="/A370079/b370079.txt">Table of n, a(n) for n = 1..10000</a>
%F A370079 a(n) = A005361(A350389(n)).
%F A370079 Multiplicative with a(p^e) = e if e is odd, and 1 if e is even.
%F A370079 a(n) = A005361(n)/A370080(n).
%F A370079 a(n) >= 1, with equality if and only if n is in A335275.
%F A370079 a(n) <= A005361(n), with equality if and only if n is an exponentially odd number (A268335).
%F A370079 Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) + 1/p^(3*s)).
%F A370079 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 1.32800597172596287374... .
%F A370079 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/((p^s - 1)*(p^s + 1)^2)). - _Vaclav Kotesovec_, Feb 11 2024
%t A370079 f[p_, e_] := If[OddQ[e], e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A370079 (PARI) a(n) = vecprod(apply(x -> if(x%2, x, 1), factor(n)[, 2]));
%Y A370079 Cf. A005361, A013661, A268335, A335275, A350387, A350389, A363329, A368472, A370080.
%K A370079 nonn,easy,mult
%O A370079 1,8
%A A370079 _Amiram Eldar_, Feb 09 2024