%I #6 May 14 2024 09:14:27
%S 7,19,21,25,56,57,59,60,62,68,71,77,79,87,175,177,179,180,186,188,189,
%T 192,193,195,196,197,204,210,212,216,218,243,244,248,254,262,263,265,
%U 279,567,572,576,577,583,592,598,599,600,602,603,605,606,610,613,616
%N 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 The binary expansion of 17 is (1,0,0,0,1) with ones minus zeros 2 - 3 = -1, and 17 is the 7th prime, 7 is in the sequence.
%e The primes A000040(a(n)) together with their binary expansions and binary indices begin:
%e 17: 10001 ~ {1,5}
%e 67: 1000011 ~ {1,2,7}
%e 73: 1001001 ~ {1,4,7}
%e 97: 1100001 ~ {1,6,7}
%e 263: 100000111 ~ {1,2,3,9}
%e 269: 100001101 ~ {1,3,4,9}
%e 277: 100010101 ~ {1,3,5,9}
%e 281: 100011001 ~ {1,4,5,9}
%e 293: 100100101 ~ {1,3,6,9}
%e 337: 101010001 ~ {1,5,7,9}
%e 353: 101100001 ~ {1,6,7,9}
%e 389: 110000101 ~ {1,3,8,9}
%e 401: 110010001 ~ {1,5,8,9}
%e 449: 111000001 ~ {1,7,8,9}
%e 1039: 10000001111 ~ {1,2,3,4,11}
%e 1051: 10000011011 ~ {1,2,4,5,11}
%e 1063: 10000100111 ~ {1,2,3,6,11}
%e 1069: 10000101101 ~ {1,3,4,6,11}
%e 1109: 10001010101 ~ {1,3,5,7,11}
%e 1123: 10001100011 ~ {1,2,6,7,11}
%e 1129: 10001101001 ~ {1,4,6,7,11}
%e 1163: 10010001011 ~ {1,2,4,8,11}
%t Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==-1&]
%Y Restriction of A031444 (positions of '-1's in A145037) to A000040.
%Y Taking primes gives A095072.
%Y Positions of negative ones in A372516, absolute value A177718.
%Y The negative version is A372538, taking primes A095073.
%Y A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A030190 gives binary expansion, reversed A030308.
%Y A035103 counts zeros in binary expansion of primes, firsts A372474.
%Y A048793 lists binary indices, reverse A272020, sum A029931.
%Y A070939 gives the length of an integer's binary expansion.
%Y A101211 lists run-lengths in binary expansion, row-lengths A069010.
%Y A372471 lists binary indices of primes.
%Y Cf. A003714, A031448, A035100, A037861, A053738, A061712, A066195, A104080, A211997, A372429, A372517, A372686.
%K nonn,base
%O 1,1
%A _Gus Wiseman_, May 14 2024