A329243 a(n) = Pi(8,3)(prime(n)) + Pi(8,5)(prime(n)) + Pi(8,7)(prime(n)) - 3*Pi(8,1)(prime(n)), where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

0, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 9, 10, 11, 8, 9, 10, 7, 4, 5, 6, 7, 8, 5, 6, 7, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 16, 17, 14, 15, 12, 13, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 12, 13, 14, 11, 12, 13, 10, 11, 12, 13, 14, 15, 16

Offset: 1

Author

Jianing Song, Nov 08 2019

Keywords

• sign

Comments

The initial terms are nonnegative integers, a(n) is negative for some prime(n) ~ 10^28.127. See the comments about "Chebyshev's bias" in A329242.

Examples

			For prime(25) = 97, there are 5 primes <= 97 that are congruent to 1 mod 8 (17, 41, 73, 89, 97), 7 primes congruent to 3 mod 8 (3, 11, 19, 43, 59, 67, 83), 6 primes congruent to 5 mod 8 (5, 13, 29, 37, 53, 61), 6 primes congruent to 7 mod 8 (7, 23, 31, 47, 71, 79), so a(25) = 7 + 6 + 6 - 3*5 = 4.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A329242.

Cf. A321856, A066520, A321857, A321859, A321860.

Programs

PARI

a(n) = my(k=0); forprime(p=3, prime(n), if(p%8==1, k-=3, k++)); k

Extensions

Edited by Peter Munn, Nov 19 2023