A073310 -id:A073310 - cp's OEIS frontend

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.

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

T. D. Noe, Aug 02 2002

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

			a(4) = 5 because d(1)=3 and d(2)=3 and d(3)=5.

R. K. Guy, Unsolved Problems in Number Theory, Third Ed., Springer, 2004.

T. D. Noe, Table of n, a(n) for n=2..10000

Cf. A073310.

maxN=200; lst={}; For[n=2, n<=maxN, n++, For[soln={}; j=1, j2n, Print["Failure at n = ", n]]]; AppendTo[lst, Max[soln]]]; lst

cp's OEIS Frontend

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.

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