cp's OEIS Frontend

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

%I A327777 #16 Dec 13 2019 18:50:37
%S A327777 2,257,8519971,36574494881,140739702949921,140773995710729,
%T A327777 140774004099109
%N A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.
%C A327777 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.
%C A327777 Conjecture: This sequence is infinite.
%H A327777 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A327777 The initial terms together with their binary indices:
%e A327777                 2: {2}
%e A327777               257: {1,9}
%e A327777           8519971: {1,2,6,9,18,24}
%e A327777       36574494881: {1,6,8,16,18,27,32,36}
%e A327777   140739702949921: {1,6,12,27,32,48}
%e A327777   140773995710729: {1,4,9,12,18,32,36,48}
%e A327777   140774004099109: {1,3,6,12,18,24,32,36,48}
%t A327777 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327777 Select[Prime[Range[1000]],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]
%Y A327777 A subset of A327368.
%Y A327777 The binary weight of prime(n) is A014499(n), with binary length A035100(n).
%Y A327777 Heinz numbers of partitions with integer mean: A316413.
%Y A327777 Heinz numbers of partitions with integer geometric mean: A326623.
%Y A327777 Heinz numbers with both: A326645.
%Y A327777 Subsets with integer mean: A051293
%Y A327777 Subsets with integer geometric mean: A326027
%Y A327777 Subsets with both: A326643
%Y A327777 Partitions with integer mean: A067538
%Y A327777 Partitions with integer geometric mean: A067539
%Y A327777 Partitions with both: A326641
%Y A327777 Strict partitions with integer mean: A102627
%Y A327777 Strict partitions with integer geometric mean: A326625
%Y A327777 Strict partitions with both: A326029
%Y A327777 Factorizations with integer mean: A326622
%Y A327777 Factorizations with integer geometric mean: A326028
%Y A327777 Factorizations with both: A326647
%Y A327777 Numbers whose binary indices have integer mean: A326669
%Y A327777 Numbers whose binary indices have integer geometric mean: A326673
%Y A327777 Numbers whose binary indices have both: A327368
%Y A327777 Cf. A000120, A029931, A034797, A048793, A070939, A096111, A291166, A326031, A326644, A326699/A326700.
%K A327777 nonn,more
%O A327777 1,1
%A A327777 _Gus Wiseman_, Sep 27 2019
%E A327777 a(4)-a(7) from _Giovanni Resta_, Dec 01 2019