A095691 Multiplicative with a(p^e) = A000720(e)+1.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 2, 4, 1, 1, 1, 3, 1

Offset: 1

Author

Vladeta Jovovic, Jul 06 2004

Keywords

• mult
• nonn
• easy

Comments

The number of divisors of n that are terms of A056166. - Amiram Eldar, Oct 31 2023

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000720, A056166, A095683.

Programs

Mathematica

Array[Times @@ Map[PrimePi@# + 1 &, FactorInteger[#][[All, -1]] ] &, 120] (* Michael De Vlieger, Jul 19 2017 *)

PARI

A095691(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= (1+primepi(f[k, 2])); ); m; } \\ Antti Karttunen, Jul 19 2017

Python

from sympy import factorint, primepi, prod
def a(n): return 1 if n==1 else prod(primepi(e) + 1 for e in factorint(n).values())
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{q prime} 1/p^q) = Sum_{n>=1} 1/A056166(n) = 1.80728269690724154161... . - Amiram Eldar, Oct 31 2023