A258811 Least prime p with p+2 prime such that n = (q+1)/(p+1) for some prime q with q+2 also prime.

3, 5, 3, 17, 5, 11, 5, 29, 11, 5, 17, 5, 149, 29, 3, 11, 5, 3, 11, 11, 41, 29, 5, 17, 5, 11, 3, 269, 11, 5, 41, 5, 5, 29, 11, 11, 179, 5, 59, 5, 29, 149, 29, 29, 3, 17, 5, 3, 17, 11, 41, 5, 149, 29, 11, 59, 3, 5, 17, 3, 461, 179, 1229, 17, 29, 107, 59, 179, 11, 5

Offset: 1

Author

Zhi-Wei Sun, Jun 11 2015

Keywords

• nonn

Comments

Conjecture: a(n) does not exceed n^2-n+5. Also, the set {(p+1)/(q+1): p, q, p+2 and q+2 are all prime} contains all positive rational numbers.

Clearly, this conjecture implies the Twin Prime Conjecture.

Examples

			a(1) = 3 since 1 = (3+1)/(3+1) with 3 and 5 twin prime.
a(4) = 17 since 4 = (71+1)/(17+1) with {17,19} and {71,73} twin prime pairs.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001359, A006512, A258803.

Programs

Mathematica

TW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
Do[k=0;Label[bb];k=k+1;If[PrimeQ[Prime[k]+2]&&TW[n*(Prime[k]+1)],Goto[aa],Goto[bb]];
Label[aa];Print[n," ",Prime[k]];Continue,{n,1,70}]