A367805 a(1) = 0; for n > 1, a(n) is the least positive integer k for which k*prime(n) + 2 is prime.

0, 1, 1, 3, 1, 3, 1, 3, 3, 1, 5, 3, 1, 3, 7, 7, 1, 5, 5, 1, 5, 3, 3, 3, 3, 1, 3, 1, 5, 9, 3, 7, 1, 3, 1, 5, 5, 3, 3, 3, 1, 5, 1, 5, 1, 3, 9, 5, 1, 9, 3, 1, 15, 7, 3, 15, 1, 9, 11, 1, 9, 3, 21, 1, 3, 3, 5, 3, 1, 3, 3, 15, 3, 5, 9, 3, 13, 3, 19, 3, 1, 15, 1, 3, 3, 9, 13, 3, 1, 15

Offset: 1

Author

Frank Hollstein, Dec 01 2023

Keywords

• nonn

Examples

			For n = 4: a(4) = 3, because prime(4) = 7, 3*7 + 2 = 23 which is prime.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A029707, A035096, A117673, A279756, A001359.

Programs

Maple

a:= proc(n) local p, q, r; p:= ithprime(n); q:= p;
      while irem(q-2, p, 'r')<>0 do q:= nextprime(q) od; r
    end:
seq(a(n), n=1..99);  # Alois P. Heinz, Dec 04 2023

Mathematica

nmax=90; a[1]=0; For[n=2, n<=nmax, n++, For[k=1, k>0, k++, If[PrimeQ[k*Prime[n]+2], a[n]=k; k=-1]]]; Array[a,nmax] (* Stefano Spezia, Dec 04 2023 *)

PARI

a(n) = if (n==1, 0, my(k=1, p=prime(n)); while (!isprime(k*p+2), k++); k); \\ Michel Marcus, Dec 02 2023

Python

from itertools import count, dropwhile
from sympy import prime, isprime
def A367805(n):
    if n==1:
        return 0
    else:
        p = prime(n)
        return next(dropwhile(lambda x:not isprime(x*p+2),count(1))) # Chai Wah Wu, Jan 04 2024

Formula

a(n) = (A279756(n) - 2)/A000040(n).

a(n) = 1 <=> n in A029707.

Extensions

More terms from Michel Marcus, Dec 02 2023