A082467 Least k>0 such that n-k and n+k are both primes.

1, 2, 1, 4, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 12, 3, 2, 9, 6, 5, 6, 3, 4, 9, 12, 1, 12, 9, 4, 3, 6, 5, 6, 9, 2, 3, 12, 1, 24, 3, 2, 15, 6, 5, 12, 3, 8, 9, 6, 7, 12, 3, 4, 15, 12, 1, 18, 9, 4, 3, 6, 5, 6, 15, 2, 3, 12, 1, 6, 15, 4, 3, 6, 5, 18, 9, 2, 15, 24, 5, 12, 3, 14, 9, 18, 7, 12, 9, 4, 15, 6, 7, 30, 9

Offset: 4

Author

Benoit Cloitre, Apr 27 2003

Keywords

• nonn

Comments

The existence of k>0 for all n >= 4 is equivalent to the strong Goldbach Conjecture that every even number >= 8 is the sum of two distinct primes.

n and k are coprime, because otherwise n + k would be composite. So the rational sequence r(n) = a(n)/n = k/n is injective. - Jason Kimberley, Sep 21 2011

Because there are arbitrarily many composites from m!+2 to m!+m, there are also arbitrarily large a(n) but they increase very slowly. The twin prime conjecture implies that infinitely many a(n) are 1. - Juhani Heino, Apr 09 2020

Examples

			n=10: k=3 because 10-3 and 10+3 are both prime and 3 is the smallest k such that n +/- k are both prime.
		

Links

• Klaus Brockhaus, Table of n, a(n) for n = 4..5000
• OEIS (Plot 2), log_10(A082467(n)/n) vs n
• J. S. Kimberley, A082467

Crossrefs

Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683, A087711.

Cf. A129301 (records), A129302 (where records occur).

Cf. A047160 (allows k=0).

Cf. A078611 (subset for prime n).

Programs

Magma

A082467 := func; [A082467(n):n in [4..98]]; // Jason Kimberley, Sep 03 2011

Maple

A082467 := proc(n) local k; k := 1+irem(n,2);
while n > k do if isprime(n-k) then if isprime(n+k)
then RETURN(k) fi fi; k := k+2 od; print("Goldbach erred!") end:
seq(A082467(i),i=4..90); # Peter Luschny, Sep 21 2011

Mathematica

f[n_] := Block[{k}, If[OddQ[n], k = 2, k = 1]; While[ !PrimeQ[n - k] || !PrimeQ[n + k], k += 2]; k]; Table[ f[n], {n, 4, 98}] (* Robert G. Wilson v, Mar 28 2005 *)

PARI

a(n)=if(n<0,0,k=1; while(isprime(n-k)*isprime(n+k) == 0,k++); k)

Formula

A078496(n)-a(n) = A078587(n)+a(n) = n.

Extensions

Entries checked by Klaus Brockhaus, Apr 08 2007