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A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

%I A357490 #5 Oct 17 2022 07:07:27
%S A357490 1,2,3,4,7,8,10,15,16,17,24,31,32,36,42,63,64,69,70,81,88,98,104,127,
%T A357490 128,136,170,255,256,277,278,282,292,325,326,337,344,354,360,394,418,
%U A357490 424,511,512,513,514,515,528,547,561,568,640,682,768,769,785,792,896
%N A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.
%C A357490 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%e A357490 The terms together with their corresponding compositions begin:
%e A357490    1: (1)
%e A357490    2: (2)
%e A357490    3: (1,1)
%e A357490    4: (3)
%e A357490    7: (1,1,1)
%e A357490    8: (4)
%e A357490   10: (2,2)
%e A357490   15: (1,1,1,1)
%e A357490   16: (5)
%e A357490   17: (4,1)
%e A357490   24: (1,4)
%e A357490   31: (1,1,1,1,1)
%e A357490   32: (6)
%e A357490   36: (3,3)
%e A357490   42: (2,2,2)
%e A357490   63: (1,1,1,1,1,1)
%e A357490   64: (7)
%e A357490   69: (4,2,1)
%t A357490 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357490 Select[Range[0,1000],IntegerQ[GeometricMean[stc[#]]]&]
%Y A357490 For regular mean we have A096199, counted by A271654 (partitions A067538).
%Y A357490 Subsets whose geometric mean is an integer are counted by A326027.
%Y A357490 The unordered version (partitions) is A326623, counted by A067539.
%Y A357490 The strict case is counted by A339452, partitions A326625.
%Y A357490 These compositions are counted by A357710.
%Y A357490 A078175 lists numbers whose prime factors have integer average.
%Y A357490 A320322 counts partitions whose product is a perfect power.
%Y A357490 Cf. A051293, A078174, A102627, A301987, A326622, A326624, A326028, A326567/A326568, A326641, A326645, A335405, A357184.
%K A357490 nonn
%O A357490 1,2
%A A357490 _Gus Wiseman_, Oct 16 2022