cp's OEIS Frontend

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.

%I A329243 #15 Nov 19 2023 21:16:52
%S A329243 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,
%T A329243 5,6,7,8,9,10,11,12,13,14,11,12,13,14,15,16,17,14,15,12,13,10,11,12,
%U A329243 13,14,11,12,13,14,15,12,13,14,11,12,13,10,11,12,13,14,15,16
%N 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.
%C A329243 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.
%H A329243 Jianing Song, <a href="/A329243/b329243.txt">Table of n, a(n) for n = 1..10000</a>
%e A329243 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.
%o A329243 (PARI) a(n) = my(k=0); forprime(p=3, prime(n), if(p%8==1, k-=3, k++)); k
%Y A329243 Cf. A329242.
%Y A329243 Cf. A321856, A066520, A321857, A321859, A321860.
%K A329243 sign
%O A329243 1,3
%A A329243 _Jianing Song_, Nov 08 2019
%E A329243 Edited by _Peter Munn_, Nov 19 2023