|
%I #6 Jun 11 2015 04:50:42
%S 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,
%T 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,
%U 149,29,11,59,3,5,17,3,461,179,1229,17,29,107,59,179,11,5
%N Least prime p with p+2 prime such that n = (q+1)/(p+1) for some prime q with q+2 also prime.
%C 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.
%C Clearly, this conjecture implies the Twin Prime Conjecture.
%H Zhi-Wei Sun, <a href="/A258811/b258811.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 3 since 1 = (3+1)/(3+1) with 3 and 5 twin prime.
%e a(4) = 17 since 4 = (71+1)/(17+1) with {17,19} and {71,73} twin prime pairs.
%t TW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
%t Do[k=0;Label[bb];k=k+1;If[PrimeQ[Prime[k]+2]&&TW[n*(Prime[k]+1)],Goto[aa],Goto[bb]];
%t Label[aa];Print[n," ",Prime[k]];Continue,{n,1,70}]
%Y Cf. A000040, A001359, A006512, A258803.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 11 2015
|
