A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

2, 257, 8519971, 36574494881, 140739702949921, 140773995710729, 140774004099109

Offset: 1

Author

Gus Wiseman, Sep 27 2019

Keywords

• nonn
• more

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.

Examples

			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}
		

Links

• Wikipedia, Geometric mean

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.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Prime[Range[1000]],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

Extensions

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