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A372538 Numbers k such that the number of ones minus the number of zeros in the binary expansion of the k-th prime number is 1.

%I A372538 #7 May 14 2024 09:14:32
%S A372538 3,8,20,23,24,26,30,58,61,63,65,67,78,80,81,82,84,88,185,187,194,200,
%T A372538 201,203,213,214,215,221,225,226,227,234,237,246,249,253,255,256,257,
%U A372538 259,266,270,280,284,287,290,573,578,586,588,591,593,611,614,615,626
%N A372538 Numbers k such that the number of ones minus the number of zeros in the binary expansion of the k-th prime number is 1.
%e A372538 The binary expansion of 83 is (1,0,1,0,0,1,1) with ones minus zeros 4 - 3 = 1, and 83 is the 23rd prime, so 23 is in the sequence.
%e A372538 The primes A000040(a(n)) together with their binary expansions and binary indices begin:
%e A372538      5:           101 ~ {1,3}
%e A372538     19:         10011 ~ {1,2,5}
%e A372538     71:       1000111 ~ {1,2,3,7}
%e A372538     83:       1010011 ~ {1,2,5,7}
%e A372538     89:       1011001 ~ {1,4,5,7}
%e A372538    101:       1100101 ~ {1,3,6,7}
%e A372538    113:       1110001 ~ {1,5,6,7}
%e A372538    271:     100001111 ~ {1,2,3,4,9}
%e A372538    283:     100011011 ~ {1,2,4,5,9}
%e A372538    307:     100110011 ~ {1,2,5,6,9}
%e A372538    313:     100111001 ~ {1,4,5,6,9}
%e A372538    331:     101001011 ~ {1,2,4,7,9}
%e A372538    397:     110001101 ~ {1,3,4,8,9}
%e A372538    409:     110011001 ~ {1,4,5,8,9}
%e A372538    419:     110100011 ~ {1,2,6,8,9}
%e A372538    421:     110100101 ~ {1,3,6,8,9}
%e A372538    433:     110110001 ~ {1,5,6,8,9}
%e A372538    457:     111001001 ~ {1,4,7,8,9}
%e A372538   1103:   10001001111 ~ {1,2,3,4,7,11}
%e A372538   1117:   10001011101 ~ {1,3,4,5,7,11}
%e A372538   1181:   10010011101 ~ {1,3,4,5,8,11}
%e A372538   1223:   10011000111 ~ {1,2,3,7,8,11}
%t A372538 Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==1&]
%Y A372538 Restriction of A031448 to the primes, positions of ones in A145037.
%Y A372538 Taking primes gives A095073, negative A095072.
%Y A372538 Positions of ones in A372516, absolute value A177718.
%Y A372538 Cf. A066196, A095070-A095075, A177796, A372539.
%Y A372538 A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A372538 A030190 gives binary expansion, reversed A030308.
%Y A372538 A035103 counts zeros in binary expansion of primes, firsts A372474.
%Y A372538 A048793 lists binary indices, reverse A272020, sum A029931.
%Y A372538 A070939 gives the length of an integer's binary expansion.
%Y A372538 A101211 lists run-lengths in binary expansion, row-lengths A069010.
%Y A372538 A372471 lists binary indices of primes.
%Y A372538 Cf. A000040, A003714, A035100, A037861, A053738, A061712, A066195, A104080, A211997, A372429, A372517, A372686.
%K A372538 nonn,base
%O A372538 1,1
%A A372538 _Gus Wiseman_, May 13 2024