A261282 Least positive integer k such that prime(k)*prime(k*n) = prime(p)+2 for some prime p.

14, 60, 135, 41, 199, 2, 2, 2, 61, 2, 183, 25, 15, 12, 47, 143, 110, 294, 117, 88, 22, 402, 26, 269, 116, 145, 164, 6, 10, 488, 2, 44, 120, 4, 127, 144, 119, 704, 1058, 368, 104, 2, 6, 214, 4, 129, 2, 3, 301, 2, 2, 466, 20, 107, 280, 14, 337, 12, 22, 12, 242, 1705, 415, 10, 115, 50, 2, 420, 4, 15

Offset: 1

Author

Zhi-Wei Sun, Aug 14 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(m)*prime(n) = prime(p)+2 for some prime p.

For example, 14/19 = 24528/33288, and prime(24528)*prime(33288) = 281153*392723 = 110415249619 = prime(4528436431)+2 with 4528436431 prime.

The conjecture implies that there are infinitely many primes p such that prime(p)+2 is a product of two primes. Recall that a prime p is called a Chen prime if p+2 is a product of at most two primes.

Examples

			a(2) = 60 since prime(60)*prime(60*2) = 281*659 = 185179 = prime(16763)+2 with 16763 prime.
		

References

• Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
• 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.

Crossrefs

Cf. A000040, A109611, A257926, A259487, A261281.

Programs

Mathematica

f[n_]:=Prime[n]
PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]