A071325 - OEIS

%I #26 Oct 06 2024 09:13:38

%S 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,

%T 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,

%U 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

%N Number of squares > 1 dividing n.

%H Antti Karttunen, <a href="/A071325/b071325.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

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

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

%F G.f.: Sum_{k>=2} x^(k^2)/(1 - x^(k^2)). - _Ilya Gutkovskiy_, Jan 04 2017

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

%t a[n_] := DivisorSum[n, Boole[#>1 && IntegerQ[Sqrt[#]]]&]

%t Array[a, 100] (* _Jean-François Alcover_, Dec 10 2021 *)

%o (PARI) a(n) = sumdiv(n, d, issquare(d) && (d>1)); \\ _Michel Marcus_, Jan 04 2017

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A071325(n): return prod((e>>1)+1 for e in factorint(n).values())-1 # _Chai Wah Wu_, Oct 06 2024

%Y Cf. A008966, A013661, A046951, A057427.

%K nonn

%O 1,16

%A _Reinhard Zumkeller_, May 18 2002