

A330556


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.


10



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, 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, 8, 9, 9, 9, 9, 9, 10, 11, 11, 11, 12, 12, 12, 11, 11, 12, 12, 12, 13, 13, 13
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OFFSET

0,3


COMMENTS

a(n) = A330557(n)  A330558(n).
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)?


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..99999
StackExchange, Asymptotic Distribution of Prime Gaps in Residue Classes.


EXAMPLE

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) = 31 = 2.


CROSSREFS

Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556A330561.
Sequence in context: A161072 A161111 A161046 * A341674 A230404 A082647
Adjacent sequences: A330553 A330554 A330555 * A330557 A330558 A330559


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Dec 29 2019


STATUS

approved



