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A216864
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Number of squares that divide the product of divisors of n.
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2
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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end proc:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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