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Product of the even divisors of n.
5

%I #32 Jun 26 2022 02:17:04

%S 1,2,1,8,1,12,1,64,1,20,1,576,1,28,1,1024,1,216,1,1600,1,44,1,110592,

%T 1,52,1,3136,1,3600,1,32768,1,68,1,373248,1,76,1,512000,1,7056,1,7744,

%U 1,92,1,84934656,1,1000,1,10816,1,11664,1,1404928,1,116,1,207360000,1,124,1

%N Product of the even divisors of n.

%H Harvey P. Dale, <a href="/A125911/b125911.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A007955(n) / A136655(n).

%F From _Wesley Ivan Hurt_, Jun 08 2020: (Start)

%F a(n) = Product_{d|n, d even} d.

%F If n is odd or an even squarefree number, then a(n) = floor((2*n)^(1 - ceiling(n/2) + floor(n/2)) * (d(n)/4)). (End)

%t Table[Times@@Select[Divisors[n],EvenQ],{n,70}] (* _Harvey P. Dale_, Jul 06 2017 *)

%t a[n_] := Module[{e = IntegerExponent[n, 2], o, d}, o = n/2^e; d = DivisorSigma[0, o]; n^(d*(e+1)/2)/o^(d/2)]; Array[a, 100] (* _Amiram Eldar_, Jun 26 2022 *)

%o (PARI) a(n) = my(d=divisors(n)); prod(k=1, #d, if (!(d[k]%2), d[k], 1)); \\ _Michel Marcus_, Jun 08 2020

%Y Cf. A136655, A126192.

%Y Cf. A001227, A183063, A000593, A007955.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jul 14 2008