A260252 Least prime p such that n = (prime(6*q)-1)/(prime(6*p)-1) for some prime q.

2, 18253, 3, 19, 2, 41, 43, 1087, 263, 29, 2, 281, 83, 8941, 613, 827, 7, 1867, 811, 139, 919, 13, 59, 11551, 10303, 10903, 2707, 3019, 1297, 5, 7333, 1609, 541, 701, 2281, 499, 2713, 6691, 41, 79, 1447, 1409, 263, 2129, 641, 2467, 7741, 1229, 523, 6781

Offset: 1

Author

Zhi-Wei Sun, Jul 20 2015

Keywords

• nonn

Comments

Conjecture: Let n be any positive integer, and let s and t belong to the set {1,-1}. Then each positive rational number r can be written as (prime(p*n)+s)/(prime(q*n)+t) with p and q both prime, unless n > r = 1 and {s,t} = {1,-1}.

This extends the conjecture in A258803.

For example, for n = 8, s = t = -1 and r = 16/11, we have (prime(407249*8)-1) /(prime(286411*8)-1) = 54568320/37515720 = r with 407249 and 286411 both prime. Also, for n = 10, s = -1, t = 1, and r = 23/17, we have (prime(1923029*10)-1)/(prime(1444903*10)+1) = 358404768/264907872 = r with 1923029 and 1444903 both prime.

Examples

			a(1) = 2 since 1 = (prime(6*2)-1)/(prime(6*2)-1) with 2 prime.
a(2) = 18253 since 2 = 2868672/1434336 = (prime(6*34673)-1)/(prime(6*18253)-1) with 18253 and 34673 both prime.
		

References

• Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28-Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
• Zhi-Wei Sun, Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..30
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641.

Crossrefs

Cf. A000040, A258803, A260232.

Programs

Mathematica

PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/6]
Do[k=0;Label[aa];k=k+1;If[PQ[(Prime[6*Prime[k]]-1)*n+1],Goto[bb],Goto[aa]];Label[bb];Print[n, " ", Prime[k]];Continue,{n,1,50}]