A357669 a(n) is the number of divisors of the powerful part of n.

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 4, 3, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset

1,4

Comments

The corresponding sum of divisors of the powerful part of n is A295294.

Links

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

Formula

a(n) = A000005(A057521(n)).

a(n) = A000005(n)/A056671(n).

a(n) = A000005(A064549(A003557(n))).

a(n) = 1 iff n is squarefree (A005117).

a(n) = A000005(n) iff n is powerful (A001694).

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 - p^2 + 2*p - 1)/(p^2*(p - 1))) = 2.71098009471568319328... .

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 2/p^(2*s) - 1/p^(3*s)). - Amiram Eldar, Sep 09 2023

Mathematica

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

Prog

(PARI) a(n) = {my(e = factor(n)[, 2]); prod(i=1, #e, if(e[i] == 1, 1, e[i] + 1))};

Crossrefs

Cf. A000005, A001694, A003557, A005117, A056671, A057521, A064549, A295294.

Sequence in context: A318873 A346403 A295924 * A361012 A363903 A123940

Adjacent sequences: A357666 A357667 A357668 * A357670 A357671 A357672

Keyword

nonn,easy,mult

Author

Amiram Eldar, Oct 08 2022

Status

approved