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A357490
Numbers k such that the k-th composition in standard order has integer geometric mean.
2
1, 2, 3, 4, 7, 8, 10, 15, 16, 17, 24, 31, 32, 36, 42, 63, 64, 69, 70, 81, 88, 98, 104, 127, 128, 136, 170, 255, 256, 277, 278, 282, 292, 325, 326, 337, 344, 354, 360, 394, 418, 424, 511, 512, 513, 514, 515, 528, 547, 561, 568, 640, 682, 768, 769, 785, 792, 896
OFFSET
1,2
COMMENTS
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.
EXAMPLE
The terms together with their corresponding compositions begin:
1: (1)
2: (2)
3: (1,1)
4: (3)
7: (1,1,1)
8: (4)
10: (2,2)
15: (1,1,1,1)
16: (5)
17: (4,1)
24: (1,4)
31: (1,1,1,1,1)
32: (6)
36: (3,3)
42: (2,2,2)
63: (1,1,1,1,1,1)
64: (7)
69: (4,2,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 1000], IntegerQ[GeometricMean[stc[#]]]&]
CROSSREFS
For regular mean we have A096199, counted by A271654 (partitions A067538).
Subsets whose geometric mean is an integer are counted by A326027.
The unordered version (partitions) is A326623, counted by A067539.
The strict case is counted by A339452, partitions A326625.
These compositions are counted by A357710.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Sequence in context: A065294 A240073 A332579 * A333778 A272919 A343603
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 16 2022
STATUS
approved