A372689 - OEIS

%I #6 May 19 2024 19:45:40

%S 2,3,4,6,9,11,12,16,18,23,26,29,33,38,41,43,44,48,50,55,58,61,64,69,

%T 71,72,74,79,81,86,89,91,92,96,101,103,104,106,111,113,118,121,131,

%U 132,134,137,142,144,149,151,152,154,159,163,164,166,169,174,176,181

%N Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a 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 Note the function taking a set s to its binary rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

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

%e    2:      10 ~ {2}

%e    3:      11 ~ {1,2}

%e    4:     100 ~ {3}

%e    6:     110 ~ {2,3}

%e    9:    1001 ~ {1,4}

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

%e   12:    1100 ~ {3,4}

%e   16:   10000 ~ {5}

%e   18:   10010 ~ {2,5}

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

%e   26:   11010 ~ {2,4,5}

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

%e   33:  100001 ~ {1,6}

%e   38:  100110 ~ {2,3,6}

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

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

%e   44:  101100 ~ {3,4,6}

%e   48:  110000 ~ {5,6}

%e   50:  110010 ~ {2,5,6}

%e   55:  110111 ~ {1,2,3,5,6}

%e   58:  111010 ~ {2,4,5,6}

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

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

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

%Y Union of prime-indexed rows of A118462.

%Y For even instead of prime we have A158704, odd A158705.

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

%Y The prime case is A372885, indices 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 A372471 lists binary indices of primes, row-sums A372429.

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

%Y A372689 lists numbers whose binary indices sum to a prime.

%Y A372885 lists primes whose binary indices sum to a prime, indices A372886.

%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. A035100, A071814, A096111, A023506, A277319, A372688, A372850, A372887.

%K nonn,base

%O 1,1

%A _Gus Wiseman_, May 18 2024