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Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.
7

%I #6 May 19 2024 19:42:55

%S 2,3,11,23,29,41,43,61,71,79,89,101,103,113,131,137,149,151,163,181,

%T 191,197,211,239,269,271,281,293,307,331,349,353,373,383,401,433,457,

%U 491,503,509,523,541,547,593,641,683,701,709,743,751,761,773,827,863,887

%N 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 indices of these primes are A372886.

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

%e The terms together with their binary expansions and binary indices begin:

%e 2: 10 ~ {2}

%e 3: 11 ~ {1,2}

%e 11: 1011 ~ {1,2,4}

%e 23: 10111 ~ {1,2,3,5}

%e 29: 11101 ~ {1,3,4,5}

%e 41: 101001 ~ {1,4,6}

%e 43: 101011 ~ {1,2,4,6}

%e 61: 111101 ~ {1,3,4,5,6}

%e 71: 1000111 ~ {1,2,3,7}

%e 79: 1001111 ~ {1,2,3,4,7}

%e 89: 1011001 ~ {1,4,5,7}

%e 101: 1100101 ~ {1,3,6,7}

%e 103: 1100111 ~ {1,2,3,6,7}

%e 113: 1110001 ~ {1,5,6,7}

%e 131: 10000011 ~ {1,2,8}

%e 137: 10001001 ~ {1,4,8}

%e 149: 10010101 ~ {1,3,5,8}

%e 151: 10010111 ~ {1,2,3,5,8}

%e 163: 10100011 ~ {1,2,6,8}

%e 181: 10110101 ~ {1,3,5,6,8}

%e 191: 10111111 ~ {1,2,3,4,5,6,8}

%e 197: 11000101 ~ {1,3,7,8}

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

%Y For prime instead of binary indices we have A006450, prime case of A316091.

%Y Prime numbers p such that A029931(p) is also prime.

%Y Prime case of A372689.

%Y The indices of these primes are A372886.

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

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

%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 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 Cf. A000009, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372850, A372887.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 19 2024