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A330559
a(n) = (number of primes p <= prime(n) with Delta(p) == 2 (mod 4)) - (number of primes p <= prime(n) with Delta(p) == 0 (mod 4)), where Delta(p) = nextprime(p) - p.
3
0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 15, 16, 15, 14, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 22, 23, 24, 25, 26, 25, 26, 25, 26, 27, 26, 27, 26, 25, 24, 25, 26, 27, 28, 29, 28
OFFSET
1,3
COMMENTS
Equals A330560 - A330561.
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)?
Let s = A024675, the interprimes. For each n let E(n) = number of even terms of s that are <= n, and let O(n) = number of odd terms of s that are <= n. Then a(n+1) = E(n) - O(n). That is, as we progress through s, the number of evens stays greater than the number of odds. - Clark Kimberling, Feb 26 2024
EXAMPLE
n=6: prime(6) = 13, primes p <= 13 with Delta(p) == 2 (mod 4) are 3,5,11; primes p <= 13 with Delta(p) == 0 (mod 4) are 7,13; so a(6) = 3-2 = 1.
MATHEMATICA
Join[{0}, Accumulate[Mod[Differences[Prime[Range[2, 100]]], 4] - 1]] (* Paolo Xausa, Feb 05 2024 *)
CROSSREFS
Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556, A330557, A330558, A330560, A330561.
Sequence in context: A030738 A341541 A359953 * A114228 A230287 A336336
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 30 2019
STATUS
approved