A372516 Number of ones minus number of zeros in the binary expansion of the n-th prime number.

0, 2, 1, 3, 2, 2, -1, 1, 3, 3, 5, 0, 0, 2, 4, 2, 4, 4, -1, 1, -1, 3, 1, 1, -1, 1, 3, 3, 3, 1, 7, -2, -2, 0, 0, 2, 2, 0, 2, 2, 2, 2, 6, -2, 0, 2, 2, 6, 2, 2, 2, 6, 2, 6, -5, -1, -1, 1, -1, -1, 1, -1, 1, 3, 1, 3, 1, -1, 3, 3, -1, 3, 5, 3, 5, 7, -1, 1, -1, 1, 1

Offset: 1

Author

Gus Wiseman, May 13 2024

Keywords

• sign

Comments

Absolute value is A177718.

Examples

			The binary expansion of 83 is (1,0,1,0,0,1,1), and 83 is the 23rd prime, so a(23) = 4 - 3 = 1.
		

Crossrefs

The sum instead of difference is A035100, firsts A372684 (primes A104080).

The negative version is A037861(A000040(n)).

Restriction of A145037 to the primes.

The unsigned version is A177718.

- Positions of zeros are A177796, indices of the primes A066196.

- Positions of positive terms are indices of the primes A095070.

- Positions of negative terms are indices of the primes A095071.

- Positions of negative ones are A372539, indices of the primes A095072.

- Positions of ones are A372538, indices of the primes A095073.

- Positions of nonnegative terms are indices of the primes A095074.

- Positions of nonpositive terms are indices of the primes A095075.

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 length of binary expansion.

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

A372471 lists the binary indices of each prime.

Cf. A000043, A003714, A005940, A059305, A061712, A066195, A071814, A211997, A372429, A372517, A372686.

Programs

Mathematica

Table[DigitCount[Prime[n],2,1]-DigitCount[Prime[n],2,0],{n,100}]
DigitCount[#,2,1]-DigitCount[#,2,0]&/@Prime[Range[100]] (* Harvey P. Dale, May 09 2025 *)

Formula

a(n) = A000120(A000040(n)) - A080791(A000040(n)).

a(n) = A014499(n) - A035103(n).

a(n) = A145037(A000040(n))