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A082467
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Least k>0 such that n-k and n+k are both primes.
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15
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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; internal format)
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OFFSET
| 4,2
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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
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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
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FORMULA
| A078496(n)-a(n) = A078587(n)+a(n) = n.
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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.
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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
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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}] (* from Robert G. Wilson v Mar 28 2005 *)
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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
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CROSSREFS
| Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683, A087711.
Cf. A129301 (records), A129302 (where records occur).
Cf. A047160 (allows k=0).
Sequence in context: A175702 A080079 A183201 * A106407 A023141 A072650
Adjacent sequences: A082464 A082465 A082466 * A082468 A082469 A082470
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 27 2003
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EXTENSIONS
| Entries checked by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 08 2007
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