Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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