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

1,4

Comments

First differs from A095691 at n = 512.

Links

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

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... .

Mathematica

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

Prog

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

Crossrefs

Cf. A013661, A095691, A057521, A322483.

Sequence in context: A115568 A072909 A360721 * A095691 A368105 A182426

Adjacent sequences: A365549 A365550 A365551 * A365553 A365554 A365555

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 08 2023

Status

approved