A372539 - OEIS

%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