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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)



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.


Table of n, a(n) for n=1..7.

Wikipedia, Geometric mean


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}


bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Select[Prime[Range[1000]], IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]


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




Gus Wiseman, Sep 27 2019


a(4)-a(7) from Giovanni Resta, Dec 01 2019



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