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

%I A082467 #58 Jul 03 2025 03:07:39
%S A082467 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,
%T A082467 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,
%U A082467 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
%N A082467 Least k>0 such that n-k and n+k are both primes.
%C A082467 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.
%C A082467 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
%C A082467 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
%H A082467 Klaus Brockhaus, <a href="/A082467/b082467.txt">Table of n, a(n) for n = 4..5000</a>
%H A082467 OEIS (Plot 2), <a href="/plot2a?name1=A082467&amp;name2=A000027&amp;tform1=log+base+10&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">log_10(A082467(n)/n) vs n </a>
%H A082467 J. S. Kimberley, <a href="/wiki/User:Jason_Kimberley/A082467">A082467</a>
%F A082467 A078496(n)-a(n) = A078587(n)+a(n) = n.
%e A082467 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.
%p A082467 A082467 := proc(n) local k; k := 1+irem(n,2);
%p A082467 while n > k do if isprime(n-k) then if isprime(n+k)
%p A082467 then RETURN(k) fi fi; k := k+2 od; print("Goldbach erred!") end:
%p A082467 seq(A082467(i),i=4..90); # _Peter Luschny_, Sep 21 2011
%t A082467 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 *)
%o A082467 (PARI) a(n)=if(n<0,0,k=1; while(isprime(n-k)*isprime(n+k) == 0,k++); k)
%o A082467 (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
%Y A082467 Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683, A087711.
%Y A082467 Cf. A129301 (records), A129302 (where records occur).
%Y A082467 Cf. A047160 (allows k=0).
%Y A082467 Cf. A078611 (subset for prime n).
%K A082467 nonn
%O A082467 4,2
%A A082467 _Benoit Cloitre_, Apr 27 2003
%E A082467 Entries checked by _Klaus Brockhaus_, Apr 08 2007