OFFSET
1,1
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Conjecture: This sequence is infinite.
LINKS
Wikipedia, Geometric mean
EXAMPLE
The initial terms together with their binary indices:
2: {2}
257: {1,9}
8519971: {1,2,6,9,18,24}
36574494881: {1,6,8,16,18,27,32,36}
140739702949921: {1,6,12,27,32,48}
140773995710729: {1,4,9,12,18,32,36,48}
140774004099109: {1,3,6,12,18,24,32,36,48}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Prime[Range[1000]], IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]
CROSSREFS
A subset of A327368.
Heinz numbers of partitions with integer mean: A316413.
Heinz numbers of partitions with integer geometric mean: A326623.
Heinz numbers with both: A326645.
Subsets with integer mean: A051293
Subsets with integer geometric mean: A326027
Subsets with both: A326643
Partitions with integer mean: A067538
Partitions with integer geometric mean: A067539
Partitions with both: A326641
Strict partitions with integer mean: A102627
Strict partitions with integer geometric mean: A326625
Strict partitions with both: A326029
Factorizations with integer mean: A326622
Factorizations with integer geometric mean: A326028
Factorizations with both: A326647
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Numbers whose binary indices have both: A327368
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 27 2019
EXTENSIONS
a(4)-a(7) from Giovanni Resta, Dec 01 2019
STATUS
approved