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

1,8

Comments

First differs from A363329 at n = 32.

Links

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

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A333844 A317933 A363329 * A318497 A202150 A093818

Adjacent sequences: A370076 A370077 A370078 * A370080 A370081 A370082

Keyword

nonn,easy,mult

Author

Amiram Eldar, Feb 09 2024

Status

approved