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

Author

Amiram Eldar, Sep 07 2019

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

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

Programs

Mathematica

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

PARI

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

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)

Extensions

Name corrected by Amiram Eldar, Sep 08 2023