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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
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 (list; graph; refs; listen; history; text; internal format)
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 A094466

Adjacent sequences:  A258808 A258809 A258810 * A258812 A258813 A258814

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jun 11 2015

STATUS

approved

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Last modified April 7 13:36 EDT 2020. Contains 333305 sequences. (Running on oeis4.)