A300817 Smallest prime p such that p + n^2 is prime, or 0 if no such prime exists.

2, 2, 3, 2, 3, 0, 5, 0, 3, 2, 3, 0, 5, 0, 3, 2, 7, 0, 7, 0, 19, 2, 3, 0, 11, 0, 7, 0, 3, 0, 7, 0, 7, 2, 7, 0, 5, 0, 3, 2, 7, 0, 13, 0, 13, 2, 13, 0, 5, 0, 3, 0, 3, 0, 11, 0, 31, 2, 7, 0, 7, 0, 3, 0, 3, 0, 7, 0, 13, 0, 3, 0, 5, 0, 3, 0, 3, 0, 5, 0, 73, 2, 13, 0, 13, 0, 37, 0, 13, 0

Offset: 0

Author

Bruno Berselli, Mar 13 2018

Keywords

• nonn

Comments

a(n) = 0 if n is a member of A106571.

Examples

			For n = 16:
2 + 16^2 is not prime;
3 + 16^2 = 7*37 is not prime;
5 + 16^2 = 3*87 is not prime;
7 + 16^2 = 263 is prime, therefore a(16) = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A087242: smallest prime p such that p + n is prime.

Cf. A174960: smallest prime p such that p + n*(n+1)/2 is prime.

Cf. A106571.

Programs

Julia

using Primes
function A300817(n) p, q = 2, n * n
    n % 2 == 1 && return isprime(p + q) ? 2 : 0
    while !isprime(p + q) p = nextprime(p + 1) end
p end
[A300817(n) for n in 0:89] |> println # Peter Luschny, Mar 13 2018

Maple

A300817 := proc(n) local p, n2; p := 2; n2 := n^2;
    if irem(n2, 2) = 1 and numtheory:-invphi(n2+1) = [] then return 0 fi;
    do if isprime(p + n2) then return p fi; p := nextprime(p) od;
end: seq(A300817(n), n = 0..89); # Peter Luschny, Mar 13 2018

Mathematica

a[n_] := Block[{p=2}, If[OddQ[n], If[PrimeQ[n^2 + 2], 2, 0], While[! PrimeQ[n^2 + p], p = NextPrime[p]]; p]]; a /@ Range[0, 89] (* Giovanni Resta, Mar 13 2018 *)

PARI

A300817(n)={if(bittest(n,0), n=n^2; forprime(p=2,, isprime(2+n)&&return(p)), isprime(2+n^2)*2)} \\ M. F. Hasler, Mar 14 2018

Python

from sympy import nextprime, isprime
def A300817(n):
    p, n2 = 2, n**2
    if n % 2:
        return 2 if isprime(2+n2) else 0
    while not isprime(p+n2):
        p = nextprime(p)
    return p # Chai Wah Wu, Mar 14 2018