A087243 a(n) = n + A087242(n) or a(n)=0 if A087242(n)=0; the primes arising as n + A087242(n).

3, 5, 5, 7, 7, 11, 0, 11, 11, 13, 13, 17, 0, 17, 17, 19, 19, 23, 0, 23, 23, 29, 0, 29, 0, 29, 29, 31, 31, 37, 0, 37, 0, 37, 37, 41, 0, 41, 41, 43, 43, 47, 0, 47, 47, 53, 0, 53, 0, 53, 53, 59, 0, 59, 0, 59, 59, 61, 61, 67, 0, 67, 0, 67, 67, 71, 0, 71, 71, 73, 73, 79, 0, 79, 0, 79, 79, 83, 0

Offset: 1

Author

Labos Elemer, Sep 04 2003

Keywords

• nonn

Comments

If n is odd, a(n) is either n+2 (if n+2 is prime) or 0; if n is even, a(n) is the least prime p such that p-n is prime. Polignac's conjecture implies that such a prime exists. - Robert Israel, Sep 29 2014

Examples

			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+(oddprime) 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.
		

Crossrefs

Cf. A087242.

Programs

Maple

N:= 1000: # to get the first N terms
nToDo:= floor(N/2): OddPrimes[1]:= 3: A[1]:= 3:
for i from 1 to floor(N/2) do A[2*i+1]:= 0 od:
for j from 2 while nToDo > 0 do
   OddPrimes[j]:= nextprime(OddPrimes[j-1]);
   A[OddPrimes[j]-2]:= OddPrimes[j];
   for i from 1 to j-1 do
      d:= OddPrimes[j] - OddPrimes[i];
      if d <= N and not assigned(A[d]) then
         A[d]:= OddPrimes[j];
         nToDo:= nToDo-1;
      fi
    od
od:
seq(A[j], j=1..N); # Robert Israel, Sep 29 2014

PARI

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

Formula

a(n) = n+Min{x prime; n+x is prime} or a(n)=0 if Min{} does not exist.

Extensions

Some corrections by Michel Marcus, Dec 26 2013