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

1,1

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A046882 A165812 A218405 * A074318 A276718 A102200

Adjacent sequences: A259489 A259490 A259491 * A259493 A259494 A259495

Keyword

nonn

Author

Zhi-Wei Sun, Jun 28 2015

Status

approved