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Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.
3

%I #9 Oct 31 2022 07:32:32

%S 1,1,1,1,3,1,7,1,1,5,1,1,9,7,3,1,3,1,7,11,13,1,7,1,11,35,25,31,27,5,

%T 157,1,31,131,39,31,33,37,183,179,135,157,7,265,3,871,187,865,259,879,

%U 867,179,1593,6073,1593,271,5995,149,6661,2411,1509,997,1045,5887

%N Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeometricMean.html">Geometric Mean</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%e a(10) = 5 because we have [10], [9, 1], [1, 9], [8, 2] and [2, 8].

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,15}] (* _Gus Wiseman_, Oct 30 2022 *)

%Y For partitions we have A326625, non-strict A067539 (ranked by A326623).

%Y The version for subsets is A326027.

%Y For arithmetic mean we have A339175, non-strict A271654.

%Y The non-strict case is counted by A357710, ranked by A357490.

%Y A032020 counts strict compositions.

%Y A067538 counts partitions with integer average.

%Y A078175 lists numbers whose prime factors have integer average.

%Y A320322 counts partitions whose product is a perfect power.

%Y Cf. A051293, A078174, A096199, A102627, A326622, A326624, A326028, A326641, A326645, A335405.

%K nonn

%O 1,5

%A _Ilya Gutkovskiy_, Dec 05 2020