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

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, 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, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Feb 09 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A363329 at n = 32.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005361, A013661, A268335, A335275, A350387, A350389, A363329, A368472, A370080.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> if(x%2, x, 1), factor(n)[, 2]));

Formula

a(n) = A005361(A350389(n)).

Multiplicative with a(p^e) = e if e is odd, and 1 if e is even.

a(n) = A005361(n)/A370080(n).

a(n) >= 1, with equality if and only if n is in A335275.

a(n) <= A005361(n), with equality if and only if n is an exponentially odd number (A268335).

Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) + 1/p^(3*s)).

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... .

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/((p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024