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A073316
a(n) = Max d(j), j=1..n-1, where d(j) is the smallest positive number such that 2j+d(j) and 2n+d(j) are both prime. A generalization of A073310.
2
1, 1, 5, 3, 5, 5, 3, 5, 11, 9, 17, 15, 13, 17, 15, 13, 11, 23, 21, 19, 23, 21, 23, 21, 19, 17, 15, 13, 23, 21, 19, 17, 15, 13, 29, 29, 27, 25, 23, 21, 19, 17, 15, 23, 21, 19, 17, 15, 13, 29, 35, 33, 31, 41, 39, 53, 51, 49, 47, 45, 43, 41, 39, 37, 35, 33, 31, 35, 33, 31, 29, 27
OFFSET
2,3
COMMENTS
Conjecture: a(n) < 2n. Note that the truth of this conjecture implies that for any pair of positive even numbers e1 < e2 <= 2n, there is a positive odd number d < 2n such that e1+d and e2+d are primes. Note that this conjecture can also be stated with odd and even swapped: for any pair of positive odd numbers d1 < d2 < 2n, there is a positive even number e <= 2n such that e+d1 and e+d2 are primes. Also note that proving this conjecture would prove the twin primes conjecture.
This is equivalent to a conjecture by Erdos mentioned by R. K. Guy at the end of section C1 of his book. The conjecture has been verified for n < 10^5. - T. D. Noe, Nov 04 2007
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, Third Ed., Springer, 2004.
EXAMPLE
a(4) = 5 because d(1)=3 and d(2)=3 and d(3)=5.
MATHEMATICA
maxN=200; lst={}; For[n=2, n<=maxN, n++, For[soln={}; j=1, j<n, j++, k=1; While[k<2n&&!(PrimeQ[k+2n]&&PrimeQ[k+2j]), k=k+2]; AppendTo[soln, k]; If[k>2n, Print["Failure at n = ", n]]]; AppendTo[lst, Max[soln]]]; lst
CROSSREFS
Cf. A073310.
Sequence in context: A165096 A165098 A194584 * A245979 A277621 A228046
KEYWORD
easy,nonn
AUTHOR
T. D. Noe, Aug 02 2002
STATUS
approved