A326647 - OEIS

%I #6 Jul 18 2019 06:18:18

%S 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 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 1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>

%e The a(216) = 5 factorizations:

%e   (2*4*27)

%e   (3*3*24)

%e   (3*6*12)

%e   (6*6*6)

%e   (216)

%e The a(729) = 8 factorizations:

%e   (3*3*3*3*3*3)

%e   (3*3*81)

%e   (3*9*27)

%e   (3*243)

%e   (9*9*9)

%e   (9*81)

%e   (27*27)

%e   (729)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,2,100}]

%Y Positions of terms > 1 are the perfect powers A001597.

%Y Factorizations with integer average are A326622.

%Y Factorizations with integer geometric mean are A326028.

%Y Partitions with integer average and geometric mean are A326641.

%Y Cf. A001055, A067538, A067539, A322794, A326514, A326515, A326516, A326623, A326643, A326645.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jul 16 2019