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%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