A365552 The number of exponentially odd divisors of the powerful part of n.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Sep 08 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A095691 at n = 512.

Links

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

Crossrefs

Cf. A013661, A095691, A057521, A322483.

Programs

Mathematica

f[p_, e_] := If[e == 1, 1, Floor[(e + 3)/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> if(x == 1, 1, (x+3)\2), factor(n)[, 2]));

Formula

a(n) = A322483(A057521(n)).

Multiplicative with a(p) = 1 and a(p^e) = floor((e+3)/2) for e >= 2.

Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(4*s)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4) = 1.80989829762278336163... .