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A082467 Least k>0 such that n-k and n+k are both primes. 25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

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 03 and 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

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

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

(Magma) A082467 := func<n|exists(r){m:m in[1..n-2]|IsPrime(n-m) and IsPrime(n+m)} select r else-1>; [A082467(n):n in [4..98]]; // Jason Kimberley, Sep 03 2011

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).

Sequence in context: A277749 A227629 A183201 * A106407 A023141 A283324

Adjacent sequences:  A082464 A082465 A082466 * A082468 A082469 A082470

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Apr 27 2003

EXTENSIONS

Entries checked by Klaus Brockhaus, Apr 08 2007

STATUS

approved

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Last modified November 26 05:48 EST 2022. Contains 358353 sequences. (Running on oeis4.)