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A326647 Number of factorizations of n into factors > 1 with integer average and integer geometric mean.

%I A326647 #6 Jul 18 2019 06:18:18
%S A326647 0,1,1,2,1,1,1,2,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,
%T A326647 1,3,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,
%U A326647 1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1
%N A326647 Number of factorizations of n into factors > 1 with integer average and integer geometric mean.
%H A326647 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A326647 The a(216) = 5 factorizations:
%e A326647   (2*4*27)
%e A326647   (3*3*24)
%e A326647   (3*6*12)
%e A326647   (6*6*6)
%e A326647   (216)
%e A326647 The a(729) = 8 factorizations:
%e A326647   (3*3*3*3*3*3)
%e A326647   (3*3*81)
%e A326647   (3*9*27)
%e A326647   (3*243)
%e A326647   (9*9*9)
%e A326647   (9*81)
%e A326647   (27*27)
%e A326647   (729)
%t A326647 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A326647 Table[Length[Select[facs[n],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,2,100}]
%Y A326647 Positions of terms > 1 are the perfect powers A001597.
%Y A326647 Factorizations with integer average are A326622.
%Y A326647 Factorizations with integer geometric mean are A326028.
%Y A326647 Partitions with integer average and geometric mean are A326641.
%Y A326647 Cf. A001055, A067538, A067539, A322794, A326514, A326515, A326516, A326623, A326643, A326645.
%K A326647 nonn
%O A326647 1,4
%A A326647 _Gus Wiseman_, Jul 16 2019