A259492 Least positive integer k such that prime(k)-k, prime(k)+k, prime(k*n)-k*n, prime(k*n)+k*n, prime(k)+k*n and prime(k*n)+k are all prime.

4, 48852, 6, 27330, 89814, 13080, 9570, 44592, 6762, 28560, 1560, 8580, 2958, 672, 9816, 6300, 40050, 53580, 3354, 858, 4530, 100650, 182520, 49740, 48660, 25296, 66990, 87120, 43680, 6840, 52122, 2970, 22770, 15888, 34704, 406350, 67890, 99630, 92490, 83064, 28614, 8580, 32070, 42, 50442, 38676, 818202, 30450, 47880, 4620

Offset: 1

Author

Zhi-Wei Sun, Jun 28 2015

Keywords

• nonn

Comments

Conjecture: Any positive rational number r can be written as m/n with prime(m)-m, prime(m)+m, prime(n)-n, prime(n)+n, prime(m)+n and m+prime(n) all prime.

Examples

			a(3) = 6 since prime(6)-6 = 7, prime(6)+6 = 19, prime(6*3)-6*3 = 43, prime(6*3)+6*3 = 79, prime(6)+6*3 = 31 and prime(6*3)+6 = 67 are all 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..500
• Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..150
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A258803, A258836, A259487, A259540.

Programs

Mathematica

PQ[k_]:=PrimeQ[Prime[k]-k]&&PrimeQ[Prime[k]+k]
QQ[m_,n_]:=PQ[m]&&PQ[n]&&PrimeQ[Prime[m]+n]&&PrimeQ[m+Prime[n]]
Do[k=0;Label[bb];k=k+1;If[QQ[k,n*k], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k];Continue,{n,1,50}]