A366147 The number of divisors of the cubefree part of n (A360539).

1, 2, 2, 3, 2, 4, 2, 1, 3, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 2, 3, 4, 1, 6, 2, 8, 2, 1, 4, 4, 4, 9, 2, 4, 4, 2, 2, 8, 2, 6, 6, 4, 2, 2, 3, 6, 4, 6, 2, 2, 4, 2, 4, 4, 2, 12, 2, 4, 6, 1, 4, 8, 2, 6, 4, 8, 2, 3, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Oct 01 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000005, A004709, A036966, A056671, A360539, A366077, A366078, A366145, A366148.

Programs

Mathematica

f[p_, e_] := If[e < 3, e+1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Formula

a(n) = A000005(A360539(n)).

a(n) = A000005(n)/A366145(n).

Multiplicative with a(p^e) = e+1 if e <= 2, and 1 otherwise.

a(n) >= 1, with equality if and only if n is cubefull (A036966).

a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).

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

From Vaclav Kotesovec, Oct 01 2023: (Start)

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

Let f(s) = Product_{p prime} (1 - 3/p^(3*s) + 2/p^(4*s)).

Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - 3/p^3 + 2/p^4) = 0.66219033176371496870504912254207846719824904470940603905284774924086...,

f'(1) = f(1) * Sum_{p prime} (9*p - 8) * log(p) / (p^4 - 3*p + 2) = f(1) * 1.04316863044761953555286128194165251303791613504188623828521117799260...

and gamma is the Euler-Mascheroni constant A001620. (End)