A259540 Least positive integer k such that k and k*n are terms of A259539.

60, 326940, 728700, 115020, 375258, 70920, 33150, 297990, 2340, 72870, 858, 1416210, 284130, 78978, 91368, 9438, 5547000, 767760, 1182918, 30468, 485208, 60, 7908810, 916188, 21522, 823968, 87720, 390210, 3252, 72870, 7878, 1823010, 1179990, 98010, 3462, 7878, 280590, 6870, 60, 434460

Offset

1,1

Comments

Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259539.

For example, 4/5 = 11673840/14592300 with 11673840 and 14592300 terms of A259539.

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 r = a/b with a,b = 1..100

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Example

a(22) = 60 since 60 and 60*22 = 1320 are terms of A259539. In fact, 60-1 = 59, 60+1 = 61, prime(60)+2 = 283, 1320-1 = 1319, 1320+1 = 1321 and prime(1320)+2 = 10861 are all prime.

Mathematica

PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]
QQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]+2]
Do[k=0; Label[bb]; k=k+1; If[PQ[k]&&QQ[n*(Prime[k]+1)], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]+1]; Continue, {n, 1, 40}]

Crossrefs

Cf. A000040, A014574, A258803, A258836, A259487, A259492, A259539.

Sequence in context: A166384 A180261 A348819 * A003795 A003921 A003928

Adjacent sequences: A259537 A259538 A259539 * A259541 A259542 A259543

Keyword

nonn

Author

Zhi-Wei Sun, Jun 30 2015

Status

approved