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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
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
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. 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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Apr 27 2003
EXTENSIONS
Entries checked by Klaus Brockhaus, Apr 08 2007
STATUS
approved