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Indices of prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.
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%I #5 May 19 2024 19:42:50

%S 1,2,5,9,10,13,14,18,20,22,24,26,27,30,32,33,35,36,38,42,43,45,47,52,

%T 57,58,60,62,63,67,70,71,74,76,79,84,88,94,96,97,99,100,101,108,116,

%U 124,126,127,132,133,135,137,144,150,154,156,160,161,162,164,172

%N Indices of prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

%C 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 The prime numbers themselves are A372885(n).

%e The binary indices of 89 = prime(24) are {1,4,5,7}, with sum 17, which is prime, so 24 is in the sequence.

%t Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[Prime[#],2]],1]]]&]

%Y Numbers k such that A029931(prime(k)) is prime.

%Y Indices of primes that belong to A372689.

%Y The indexed prime numbers themselves are A372885.

%Y A000040 lists the prime numbers, A014499 their binary indices

%Y A006450 lists primes of prime index, prime case of A316091.

%Y A019565 gives Heinz number of binary indices, adjoint A048675.

%Y A038499 counts partitions of prime length, strict A085756.

%Y Binary indices:

%Y - listed A048793, sum A029931

%Y - reversed A272020

%Y - opposite A371572, sum A230877

%Y - length A000120, complement A023416

%Y - min A001511, opposite A000012

%Y - max A070939, opposite A070940

%Y - complement A368494, sum A359400

%Y - opposite complement A371571, sum A359359

%Y A058698 counts partitions of prime numbers, strict A064688.

%Y A372687 counts strict partitions of prime binary rank, counted by A372851.

%Y A372688 counts partitions of prime binary rank, with Heinz numbers A277319.

%Y Cf. A029837, A158704, A158705, A035100, A372429, A372471, A372850.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 19 2024