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a(n) = (number of primes p <= 2*n+1 with Delta(p) == 2 mod 4) - (number of primes p <= 2*n+1 with Delta(p) == 0 mod 4), where Delta(p) = nextprime(p) - p.
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%I #36 Jan 01 2020 23:55:26

%S 0,1,2,1,1,2,1,1,2,1,1,2,2,2,3,4,4,4,3,3,4,3,3,4,4,4,5,5,5,6,7,7,7,6,

%T 6,7,8,8,8,7,7,8,8,8,7,7,7,7,6,6,7,6,6,7,6,6,7,7,7,7,7,7,7,6,6,7,7,7,

%U 8,9,9,9,9,9,10,11,11,11,12,12,12,11,11,12,12,12,13,13,13

%N a(n) = (number of primes p <= 2*n+1 with Delta(p) == 2 mod 4) - (number of primes p <= 2*n+1 with Delta(p) == 0 mod 4), where Delta(p) = nextprime(p) - p.

%C a(n) = A330557(n) - A330558(n).

%C Since Delta(prime(n)) grows roughly like log n, this probably changes sign infinitely often. When is the next time a(n) is zero, or the first time a(n) < 0 (if these values exist)?

%H N. J. A. Sloane, <a href="/A330556/b330556.txt">Table of n, a(n) for n = 0..99999</a>

%H StackExchange, <a href="https://math.stackexchange.com/questions/2307461/asymptotic-distribution-of-prime-gaps-in-residue-classes">Asymptotic Distribution of Prime Gaps in Residue Classes</a>.

%e n=5, 2*n+1=11: there are three primes <= 11 with Delta(p) == 2 mod 4, namely 3,5,11; and one with Delta(p) == 0 mod 4, namely 7; so a(5) = 3-1 = 2.

%Y Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556-A330561.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Dec 29 2019