A330561 a(n) = number of primes p <= prime(n) with Delta(p) == 0 (mod 4), where Delta(p) = nextprime(p) - p.

0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 16, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 25, 26, 27, 27, 27, 27, 27

Offset: 1

Author

N. J. A. Sloane, Dec 30 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A098059.

Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556, A330557, A330558, A330559, A330560.

Programs

Magma

[#[p:p in PrimesInInterval(1,NthPrime(n))|IsIntegral((NextPrime(p)-p)/4)]:n in [1..80]]; // Marius A. Burtea, Dec 31 2019

Maple

N:= 200: # for a(1)..a(N)
P:= [seq(ithprime(i),i=1..N+1)]:
Delta:= P[2..-1]-P[1..-2] mod 4:
R:= map(charfcn[0],Delta):
ListTools:-PartialSums(R); # Robert Israel, Dec 31 2019

Mathematica

Accumulate[Map[Boole[Mod[#, 4] == 0]&, Differences[Prime[Range[100]]]]] (* Paolo Xausa, Feb 05 2024 *)