

A082467


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


24



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
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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 103 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(nk) 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(nk)*isprime(n+k) == 0, k++); k)
(MAGMA) A082467 := func<nexists(r){m:m in[1..n2]IsPrime(nm) and IsPrime(n+m)} select r else1>; [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



