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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; [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 September 26 01:51 EDT 2022. Contains 356986 sequences. (Running on oeis4.)