A360097 a(n) = smallest k such that 2*n*k-1 and 2*n*k+1 are nonprimes.

13, 14, 20, 7, 5, 10, 4, 4, 8, 6, 7, 5, 1, 2, 4, 2, 1, 4, 2, 3, 17, 4, 2, 3, 1, 4, 4, 1, 2, 2, 2, 1, 8, 3, 8, 2, 4, 1, 8, 2, 3, 11, 1, 2, 10, 1, 1, 3, 4, 3, 2, 2, 4, 2, 2, 5, 3, 1, 1, 1, 1, 1, 9, 4, 2, 4, 1, 4, 3, 4, 1, 1, 1, 2, 2, 2, 1, 4, 3, 1, 2, 2, 4, 7, 1

Offset: 1

Author

Tamas Sandor Nagy, Jan 25 2023

Keywords

• nonn

Examples

			We try 1..k until the condition is met:
a(7) != 1 because 2*7*1 = 14 and 14 - 1 = 13, a prime.
a(7) != 2 because 2*7*2 = 28 and 28 + 1 = 29, a prime.
a(7) != 3 because 2*7*3 = 42 and 42 - 1 = 41 and 42 + 1 = 43, both primes.
a(7) = 4 because 2*7*4 = 56 and 56 - 1 = 55 and 56 + 1 = 57 are both nonprimes.
		

Links

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

Crossrefs

Cf. A018252, A124522.

Programs

Maple

f:= proc(n) local k;
    for k from 1 do
      if not isprime(2*n*k-1) and not isprime(2*n*k+1) then return k fi
    od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 07 2023

Mathematica

a[n_] := Module[{k = 1}, While[PrimeQ[2*n*k - 1] || PrimeQ[2*n*k + 1], k++]; k]; Array[a, 100] (* Amiram Eldar, Jan 25 2023 *)

PARI

a(n) = my(k=1); while(isprime(2*n*k-1) || isprime(2*n*k+1), k++); k; \\ Michel Marcus, Jan 25 2023

Python

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(1) if not isprime(2*n*k-1) and not isprime(2*n*k+1))
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Jan 25 2023

Extensions

More terms from Michael S. Branicky, Jan 25 2023