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.

3, 8, 20, 23, 24, 26, 30, 58, 61, 63, 65, 67, 78, 80, 81, 82, 84, 88, 185, 187, 194, 200, 201, 203, 213, 214, 215, 221, 225, 226, 227, 234, 237, 246, 249, 253, 255, 256, 257, 259, 266, 270, 280, 284, 287, 290, 573, 578, 586, 588, 591, 593, 611, 614, 615, 626

Offset: 1

Author

Gus Wiseman, May 13 2024

Keywords

• nonn
• base

Examples

			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.
The primes A000040(a(n)) together with their binary expansions and binary indices begin:
     5:           101 ~ {1,3}
    19:         10011 ~ {1,2,5}
    71:       1000111 ~ {1,2,3,7}
    83:       1010011 ~ {1,2,5,7}
    89:       1011001 ~ {1,4,5,7}
   101:       1100101 ~ {1,3,6,7}
   113:       1110001 ~ {1,5,6,7}
   271:     100001111 ~ {1,2,3,4,9}
   283:     100011011 ~ {1,2,4,5,9}
   307:     100110011 ~ {1,2,5,6,9}
   313:     100111001 ~ {1,4,5,6,9}
   331:     101001011 ~ {1,2,4,7,9}
   397:     110001101 ~ {1,3,4,8,9}
   409:     110011001 ~ {1,4,5,8,9}
   419:     110100011 ~ {1,2,6,8,9}
   421:     110100101 ~ {1,3,6,8,9}
   433:     110110001 ~ {1,5,6,8,9}
   457:     111001001 ~ {1,4,7,8,9}
  1103:   10001001111 ~ {1,2,3,4,7,11}
  1117:   10001011101 ~ {1,3,4,5,7,11}
  1181:   10010011101 ~ {1,3,4,5,8,11}
  1223:   10011000111 ~ {1,2,3,7,8,11}
		

Crossrefs

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

Taking primes gives A095073, negative A095072.

Positions of ones in A372516, absolute value A177718.

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

A000120 counts ones in binary expansion (binary weight), zeros A080791.

A030190 gives binary expansion, reversed A030308.

A035103 counts zeros in binary expansion of primes, firsts A372474.

A048793 lists binary indices, reverse A272020, sum A029931.

A070939 gives the length of an integer's binary expansion.

A101211 lists run-lengths in binary expansion, row-lengths A069010.

A372471 lists binary indices of primes.

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

Programs

Mathematica

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