A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset

1,8

Comments

First differs from A050361 at n = 64.

From Amiram Eldar, Sep 08 2023: (Start)

The number of exponentially odd divisors of n is A322483(n), and their sum is A033634(n).

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n. (End)

Also, the number of divisors of n that are cubefull exponentially odd numbers (A335988). - Amiram Eldar, Feb 11 2024

Links

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

Formula

Multiplicative with a(p^e) = floor((e+1)/2).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Sep 10 2022

a(n) = A046951(A350390(n)) (the number of squares dividing the largest exponentially odd divisor of n). - Amiram Eldar, Sep 01 2023

From Amiram Eldar, Sep 08 2023: (Start)

a(n) = A046951(A003557(n)).

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

Mathematica

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

Prog

(PARI) a(n) = vecprod(apply(x -> (x+1)\2, factor(n)[, 2])); \\ Amiram Eldar, Sep 01 2023

Crossrefs

Cf. A033634, A050361, A065487, A268335, A295316, A335988, A350390.

Cf. A003557, A005361 (number of coreful divisors), A046951, A268335.

Sequence in context: A368248 A362852 A061704 * A375359 A366902 A050361

Adjacent sequences: A325834 A325835 A325836 * A325838 A325839 A325840

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 07 2019

Extensions

Name corrected by Amiram Eldar, Sep 08 2023

Status

approved