cp's OEIS Frontend

A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.

%I A372689 #6 May 19 2024 19:45:40
%S A372689 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 A372689 71,72,74,79,81,86,89,91,92,96,101,103,104,106,111,113,118,121,131,
%U A372689 132,134,137,142,144,149,151,152,154,159,163,164,166,169,174,176,181
%N A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.
%C A372689 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 A372689 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 A372689 The terms together with their binary expansions and binary indices begin:
%e A372689    2:      10 ~ {2}
%e A372689    3:      11 ~ {1,2}
%e A372689    4:     100 ~ {3}
%e A372689    6:     110 ~ {2,3}
%e A372689    9:    1001 ~ {1,4}
%e A372689   11:    1011 ~ {1,2,4}
%e A372689   12:    1100 ~ {3,4}
%e A372689   16:   10000 ~ {5}
%e A372689   18:   10010 ~ {2,5}
%e A372689   23:   10111 ~ {1,2,3,5}
%e A372689   26:   11010 ~ {2,4,5}
%e A372689   29:   11101 ~ {1,3,4,5}
%e A372689   33:  100001 ~ {1,6}
%e A372689   38:  100110 ~ {2,3,6}
%e A372689   41:  101001 ~ {1,4,6}
%e A372689   43:  101011 ~ {1,2,4,6}
%e A372689   44:  101100 ~ {3,4,6}
%e A372689   48:  110000 ~ {5,6}
%e A372689   50:  110010 ~ {2,5,6}
%e A372689   55:  110111 ~ {1,2,3,5,6}
%e A372689   58:  111010 ~ {2,4,5,6}
%e A372689   61:  111101 ~ {1,3,4,5,6}
%t A372689 Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[#,2]],1]]]&]
%Y A372689 Numbers k such that A029931(k) is prime.
%Y A372689 Union of prime-indexed rows of A118462.
%Y A372689 For even instead of prime we have A158704, odd A158705.
%Y A372689 For prime indices instead of binary indices we have A316091.
%Y A372689 The prime case is A372885, indices A372886.
%Y A372689 A000040 lists the prime numbers, A014499 their binary indices.
%Y A372689 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372689 A058698 counts partitions of prime numbers, strict A064688.
%Y A372689 A372471 lists binary indices of primes, row-sums A372429.
%Y A372689 A372687 counts strict partitions of prime binary rank, counted by A372851.
%Y A372689 A372689 lists numbers whose binary indices sum to a prime.
%Y A372689 A372885 lists primes whose binary indices sum to a prime, indices A372886.
%Y A372689 Binary indices:
%Y A372689 - listed A048793, sum A029931
%Y A372689 - reversed A272020
%Y A372689 - opposite A371572, sum A230877
%Y A372689 - length A000120, complement A023416
%Y A372689 - min A001511, opposite A000012
%Y A372689 - max A070939, opposite A070940
%Y A372689 - complement A368494, sum A359400
%Y A372689 - opposite complement A371571, sum A359359
%Y A372689 Cf. A035100, A071814, A096111, A023506, A277319, A372688, A372850, A372887.
%K A372689 nonn,base
%O A372689 1,1
%A A372689 _Gus Wiseman_, May 18 2024