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

2,3

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 Erdős 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.

Links

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

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 A382313

Adjacent sequences: A073313 A073314 A073315 * A073317 A073318 A073319

Keyword

easy,nonn

Author

T. D. Noe, Aug 02 2002

Status

approved