A216864 - OEIS

%I #13 May 31 2015 17:00:35

%S 1,1,1,2,1,4,1,4,2,4,1,8,1,4,4,6,1,8,1,8,4,4,1,21,2,4,4,8,1,27,1,8,4,

%T 4,4,25,1,4,4,21,1,27,1,8,8,4,1,33,2,8,4,8,1,21,4,21,4,4,1,112,1,4,8,

%U 11,4,27,1,8,4,27,1,70,1,4,8,8,4,27,1,33

%N Number of squares that divide the product of divisors of n.

%H Giovanni Resta, <a href="/A216864/b216864.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A046951(A007955(n)).

%F a(n) = Product_{i=1..k} (1+floor(M*e_i/4)), for n>1, where the prime factorization of n is p_1^e_1*...*p_k^e_k and M = Product_{i=1..k}(1+e_i). - _Giovanni Resta_, May 31 2015

%e For n=28, the divisors are 1, 2, 4, 7, 14, 28. The product of these is 2^6*7^3. The sequence entry a(28) = 8 counts the squares 1, 7^2, 2^2, 2^2*7^2, 2^4, 2^4*7^2, 2^6 and 2^6*7^2, all of which divide 2^6*7^3.

%p A216864 := proc(n)

%p         A046951(A007955(n)) ;

%p end proc:

%p seq(A216864(n),n=1..80) ; # _R. J. Mathar_, Sep 18 2012

%t Table[Length[Select[Divisors[Times @@ Divisors[n]], IntegerQ[Sqrt[#]] &]], {n, 100}] (* _T. D. Noe_, Sep 18 2012 *)

%t a[n_] := Block[{e = Last /@ FactorInteger[n]}, Times @@ (1 + Floor[e * Times @@ (1 + e)/4])]; Array[a, 1000] (* _Giovanni Resta_, May 31 2015 *)

%o (PARI) a(n) = {my(d = divisors(n)); my(pd = prod(k=1, #d, d[k])); sumdiv(pd, dd, issquare(dd));} \\ _Michel Marcus_, May 31 2015

%Y Cf. A216677.

%K nonn

%O 1,4

%A _R. J. Mathar_, Sep 18 2012