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

1,16

Links

Antti Karttunen, Table of n, a(n) for n = 1..1000

Index entries for sequences computed from exponents in factorization of n.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A008966, A013661, A046951, A057427.

Sequence in context: A366073 A362412 A376657 * A064727 A343222 A112378

Adjacent sequences: A071322 A071323 A071324 * A071326 A071327 A071328

Keyword

nonn

Author

Reinhard Zumkeller, May 18 2002

Status

approved