A372538 - OEIS

%I #7 May 14 2024 09:14:32

%S 3,8,20,23,24,26,30,58,61,63,65,67,78,80,81,82,84,88,185,187,194,200,

%T 201,203,213,214,215,221,225,226,227,234,237,246,249,253,255,256,257,

%U 259,266,270,280,284,287,290,573,578,586,588,591,593,611,614,615,626

%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 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 The primes A000040(a(n)) together with their binary expansions and binary indices begin:

%e      5:           101 ~ {1,3}

%e     19:         10011 ~ {1,2,5}

%e     71:       1000111 ~ {1,2,3,7}

%e     83:       1010011 ~ {1,2,5,7}

%e     89:       1011001 ~ {1,4,5,7}

%e    101:       1100101 ~ {1,3,6,7}

%e    113:       1110001 ~ {1,5,6,7}

%e    271:     100001111 ~ {1,2,3,4,9}

%e    283:     100011011 ~ {1,2,4,5,9}

%e    307:     100110011 ~ {1,2,5,6,9}

%e    313:     100111001 ~ {1,4,5,6,9}

%e    331:     101001011 ~ {1,2,4,7,9}

%e    397:     110001101 ~ {1,3,4,8,9}

%e    409:     110011001 ~ {1,4,5,8,9}

%e    419:     110100011 ~ {1,2,6,8,9}

%e    421:     110100101 ~ {1,3,6,8,9}

%e    433:     110110001 ~ {1,5,6,8,9}

%e    457:     111001001 ~ {1,4,7,8,9}

%e   1103:   10001001111 ~ {1,2,3,4,7,11}

%e   1117:   10001011101 ~ {1,3,4,5,7,11}

%e   1181:   10010011101 ~ {1,3,4,5,8,11}

%e   1223:   10011000111 ~ {1,2,3,7,8,11}

%t Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==1&]

%Y Restriction of A031448 to the primes, positions of ones in A145037.

%Y Taking primes gives A095073, negative A095072.

%Y Positions of ones in A372516, absolute value A177718.

%Y Cf. A066196, A095070-A095075, A177796, A372539.

%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. A000040, A003714, A035100, A037861, A053738, A061712, A066195, A104080, A211997, A372429, A372517, A372686.

%K nonn,base

%O 1,1

%A _Gus Wiseman_, May 13 2024