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

1,1

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

Cf. A000040, A001359, A006512, A258803.

Sequence in context: A172003 A244801 A002586 * A066845 A014782 A348591

Adjacent sequences: A258808 A258809 A258810 * A258812 A258813 A258814

Keyword

nonn

Author

Zhi-Wei Sun, Jun 11 2015

Status

approved