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

Author

Amiram Eldar, Oct 08 2022

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

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