A190867 Count of the 3-full divisors of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 1

Author

R. J. Mathar, May 27 2011

Keywords

• nonn
• mult
• easy

Comments

a(n) is the number of divisors d of n with d an element of A036966.

This is the 3-full analog of the 2-full case A005361.

Examples

			a(16)=3 because the divisors of 16 are {1,2,4,8,16}, and three of these are 3-full: 1, 8=2^3 and 16=2^4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Aleksandar Ivić, On the asymptotic formulas for powerful numbers, Publications de l'Institut Mathématique, Vol. 23, No. 37 (1978), pp. 85-94; alternative link.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A005361, A036966, A065483.

Programs

Maple

f:= n -> convert(map(t -> max(1,t[2]-1), ifactors(n)[2]),`*`):
map(f, [$1..200]); # Robert Israel, Jul 19 2017

Mathematica

Table[Product[Max[{1, i - 1}], {i, FactorInteger[n][[All, 2]]}], {n, 1, 200}] (* Geoffrey Critzer, Feb 12 2015 *)
Table[1 + DivisorSum[n, 1 &, AllTrue[FactorInteger[#][[All, -1]], # > 2 &] &], {n, 120}] (* Michael De Vlieger, Jul 19 2017 *)

PARI

A190867(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= max(1,f[k, 2]-1); ); m; } \\ Antti Karttunen, Jul 19 2017

Python

from functools import reduce
from sympy import factorint
from operator import mul
def a(n): return 1 if n==1 else reduce(mul, [max(1, e - 1) for e in factorint(n).values()])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 19 2017

Formula

a(n) = Sum_{d|n, d in A036966} 1.

a(n) <= A005361(n).

Multiplicative with a(p^e) = max(1,e-1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p^2*(p-1))) (A065483). (Ivić, 1978). - Amiram Eldar, Jul 23 2022

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)). - Amiram Eldar, Sep 21 2023