A306863 a(n) is the number of primes between the n-th and (n+1)-st odd composite numbers.

2, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 0, 0

Offset: 1

Author

Zhandos Mambetaliyev, Mar 14 2019

Keywords

• nonn

Examples

			The first few odd composite numbers are 9, 15, 21, 25, 27, 33, 35, 39, 45, 49. Between 9 and 15, there are two primes (11 and 13); between 15 and 21 there are also two primes (17 and 19); between 21 and 25 there is only one prime (23), etc.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A071904, A000040, A001223 (differences between primes), A164510, A001097.

Programs

Mathematica

Differences@ PrimePi@ Complement[Range[3, #, 2], Prime@ Range[2, PrimePi@ #]] &@ 300 (* Michael De Vlieger, Apr 21 2019 *)
Differences[PrimePi/@Select[Range[3,301,2],CompositeQ]] (* Harvey P. Dale, Sep 16 2023 *)

PARI

{ b=9; for(i=6, 150, if(isprime(2*i-1)==0, print1(primepi(2*i-1)-primepi(b), ", "); b=2*i-1)) }

Python

from itertools import count
from sympy import primepi, isprime
def A306863(n):
    if n == 1: return 2
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return next(c for c in count(0) if not isprime(m+(c<<1)+2)) # Chai Wah Wu, Aug 02 2024