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

1,8

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.

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.

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

Example

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.

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 *)

Prog

(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 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

Crossrefs

Cf. A005361, A036966, A065483.

Sequence in context: A059895 A368328 A321167 * A355936 A326976 A117358

Adjacent sequences: A190864 A190865 A190866 * A190868 A190869 A190870

Keyword

nonn,mult,easy

Author

R. J. Mathar, May 27 2011

Status

approved