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

1,1

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A287407 A153240 A153241 * A334108 A332813 A323077

Adjacent sequences: A306860 A306861 A306862 * A306864 A306865 A306866

Keyword

nonn

Author

Zhandos Mambetaliyev, Mar 14 2019

Status

approved