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 A327777 Prime numbers whose binary indices have integer mean and integer geometric mean. 1
 2, 257, 8519971, 36574494881, 140739702949921, 140773995710729, 140774004099109 (list; graph; refs; listen; history; text; internal format)
 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. The binary weight of prime(n) is A014499(n), with binary length A035100(n). 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 Cf. A000120, A029931, A034797, A048793, A070939, A096111, A291166, A326031, A326644, A326699/A326700. Sequence in context: A006686 A100269 A258805 * A196288 A128697 A182422 Adjacent sequences:  A327774 A327775 A327776 * A327778 A327779 A327780 KEYWORD nonn,more AUTHOR Gus Wiseman, Sep 27 2019 EXTENSIONS a(4)-a(7) from Giovanni Resta, Dec 01 2019 STATUS approved

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Last modified February 23 07:11 EST 2020. Contains 332159 sequences. (Running on oeis4.)