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

0,1

Comments

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

Links

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

Example

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.

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 *)

Prog

(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
(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
(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

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.

Sequence in context: A279630 A279632 A363680 * A341417 A230140 A156220

Adjacent sequences: A300814 A300815 A300816 * A300818 A300819 A300820

Keyword

nonn

Author

Bruno Berselli, Mar 13 2018

Status

approved