A174960 -id:A174960 - cp's OEIS frontend

A087242 Smallest prime number p such that n+p = q is also a prime, or 0 if no such prime number exists.

2, 3, 2, 3, 2, 5, 0, 3, 2, 3, 2, 5, 0, 3, 2, 3, 2, 5, 0, 3, 2, 7, 0, 5, 0, 3, 2, 3, 2, 7, 0, 5, 0, 3, 2, 5, 0, 3, 2, 3, 2, 5, 0, 3, 2, 7, 0, 5, 0, 3, 2, 7, 0, 5, 0, 3, 2, 3, 2, 7, 0, 5, 0, 3, 2, 5, 0, 3, 2, 3, 2, 7, 0, 5, 0, 3, 2, 5, 0, 3, 2, 7, 0, 5, 0, 3, 2, 13, 0, 7, 0, 5, 0, 3, 2, 5, 0, 3, 2, 3, 2, 5, 0, 3, 2

Offset: 1

Labos Elemer, Sep 04 2003

nonn

			a(n)=0, i.e., no solution exists if n is a special prime, namely n is not a lesser twin prime; e.g., if n=7, then neither 7+2=9 nor 7+(odd prime) is a prime, thus no p prime exists such that 7+p is also a prime.
If n is a lesser twin prime then a(n)=2 is a solution because n+a(n) = n+2 = greater twin prime satisfying the condition.

Cf. A087243, A174960.

a(n) = {if (n % 2, if (isprime(n+2), p = 2, p = 0);, p = 2; while (!isprime(n+p), p = nextprime(p+1));); p;} \\ Michel Marcus, Dec 26 2013

a(n) = Min{x prime; n+x is prime}.

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

Original entry on oeis.org

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

Bruno Berselli, Mar 13 2018

nonn

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

			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.

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

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.

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

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

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 *)

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

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

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A087242 Smallest prime number p such that n+p = q is also a prime, or 0 if no such prime number exists.

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

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