OFFSET
1,1
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.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 11 2015
STATUS
approved