A216864 Number of squares that divide the product of divisors of n.

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

Offset

1,4

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Formula

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

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

Example

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.

Maple

A216864 := proc(n)
        A046951(A007955(n)) ;
end proc:
seq(A216864(n), n=1..80) ; # R. J. Mathar, Sep 18 2012

Mathematica

Table[Length[Select[Divisors[Times @@ Divisors[n]], IntegerQ[Sqrt[#]] &]], {n, 100}] (* T. D. Noe, Sep 18 2012 *)
a[n_] := Block[{e = Last /@ FactorInteger[n]}, Times @@ (1 + Floor[e * Times @@ (1 + e)/4])]; Array[a, 1000] (* Giovanni Resta, May 31 2015 *)

Prog

(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

Crossrefs

Cf. A216677.

Sequence in context: A193306 A053578 A168177 * A337175 A263432 A286518

Adjacent sequences: A216861 A216862 A216863 * A216865 A216866 A216867

Keyword

nonn

Author

R. J. Mathar, Sep 18 2012

Status

approved