A071325 Number of squares > 1 dividing n.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Author

Reinhard Zumkeller, May 18 2002

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A008966, A013661, A046951, A057427.

Programs

Mathematica

a[n_] := DivisorSum[n, Boole[#>1 && IntegerQ[Sqrt[#]]]&]
Array[a, 100] (* Jean-François Alcover, Dec 10 2021 *)

PARI

a(n) = sumdiv(n, d, issquare(d) && (d>1)); \\ Michel Marcus, Jan 04 2017

Python

from math import prod
from sympy import factorint
def A071325(n): return prod((e>>1)+1 for e in factorint(n).values())-1 # Chai Wah Wu, Oct 06 2024

Formula

a(n) = A046951(n) - 1.

A057427(a(n)) = 1 - A008966(n).

G.f.: Sum_{k>=2} x^(k^2)/(1 - x^(k^2)). - Ilya Gutkovskiy, Jan 04 2017

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/6 - 1. - Amiram Eldar, Sep 25 2022