A353048 a(n) = (number of primes p in the first n primes such that 2*p+1 is also prime) - (number of primes p in the first n primes such that 2*p-1 is also prime).

0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 3, 3, 4, 4, 3, 3, 3, 2, 2, 2, 3, 4, 4, 5, 5, 5, 4, 3, 3, 3, 2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 5, 5, 6, 5, 5, 5, 5, 4, 3, 3, 3, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 5

Offset

1,10

Comments

a(n) is the number of members of A005384 <= prime(n) minus the number of members of A005382 <= prime(n).

Links

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

T. Haddad, Prime Number Races, Université de Montréal master's thesis, 2020.

Robert Israel, Plot of (n, a(n)) for n = 1..10^6

Example

Of the first 5 primes there are 4 (2, 3, 5 and 11) such that 2*p+1 is prime and 3 (2, 3 and 7) such that 2*p-1 is prime, so a(5) = 4 - 3 = 1.

Maple

f:= proc(p) `if`(isprime(2*p+1), 1, 0) - `if`(isprime(2*p-1), 1, 0) end proc:
L:= map(f, [seq(ithprime(i), i=1..200)]):
ListTools:-PartialSums(L);

Mathematica

Accumulate[(Boole[PrimeQ[2*# + 1]] - Boole[PrimeQ[2*# - 1]]) & /@ Prime[Range[100]]] (* Amiram Eldar, Apr 20 2022 *)

Prog

(Python)
from itertools import accumulate
from sympy import isprime, prime, primerange
def f(p): return isprime(2*p+1) - isprime(2*p-1)
def aupton(nn): return list(accumulate(map(f, primerange(2, prime(nn)+1))))
print(aupton(99)) # Michael S. Branicky, Apr 20 2022
(PARI) a(n) = my(vp=primes(n)); sum(k=1, n, isprime(2*vp[k]+1)-isprime(2*vp[k]-1)); \\ Michel Marcus, Apr 21 2022

Crossrefs

Cf. A005382, A005384.

Sequence in context: A205217 A054635 A003137 * A006842 A299038 A273693

Adjacent sequences: A353045 A353046 A353047 * A353049 A353050 A353051

Keyword

sign

Author

J. M. Bergot and Robert Israel, Apr 20 2022

Status

approved