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.

%I A073316 #11 Mar 28 2014 02:30:11
%S A073316 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,
%T A073316 15,13,23,21,19,17,15,13,29,29,27,25,23,21,19,17,15,23,21,19,17,15,13,
%U A073316 29,35,33,31,41,39,53,51,49,47,45,43,41,39,37,35,33,31,35,33,31,29,27
%N 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.
%C A073316 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.
%C A073316 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
%D A073316 R. K. Guy, Unsolved Problems in Number Theory, Third Ed., Springer, 2004.
%H A073316 T. D. Noe, <a href="/A073316/b073316.txt">Table of n, a(n) for n=2..10000</a>
%e A073316 a(4) = 5 because d(1)=3 and d(2)=3 and d(3)=5.
%t A073316 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
%Y A073316 Cf. A073310.
%K A073316 easy,nonn
%O A073316 2,3
%A A073316 _T. D. Noe_, Aug 02 2002