A365632 The number of divisors of n that are terms of A072873.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Sep 14 2023

Keywords

• nonn
• easy
• mult

Comments

The sum of these divisors is A365633(n) and the largest of them is A327939(n).

Links

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

Crossrefs

Cf. A072873, A327939, A365633, A365634.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i,2]\f[i,1]);}

Formula

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} p^p/(p^p-1) = 1.3850602852... .

Extensions

Data corrected by Amiram Eldar, Sep 20 2023