A257926 Least positive integer k such that prime(k*n)+2 = prime(i*n)*prime(j*n) for some 0 < i < j.

6, 4, 10, 8, 451, 426, 622, 175, 1424, 500, 33, 703, 1761, 4428, 1563, 959, 8147, 7055, 5948, 250, 7517, 12706, 8405, 2948, 2610, 1949, 10424, 2214, 6722, 1963, 3335, 16382, 15687, 17591, 15073, 7818, 32202, 31169, 2248, 14899, 69955, 7580, 2393, 39295, 42352, 5884, 9367, 3630, 14090, 1305

Offset: 1

Author

Zhi-Wei Sun, Jul 14 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0.

This is much stronger than Chen's famous result that there are infinitely many Chen primes.

Examples

			a(1) = 6 since prime(6*1)+2 = 15 = 3*5 = prime(2*1)*prime(3*1).
a(3) = 10 since prime(10*3)+2 = 115 = 5*23 = prime(1*3)*prime(3*3).
a(149) = 1476387 since prime(1476387*149)+2 = 4666119529 = 8311*561439 = prime(7*149)*prime(310*149).
		

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..200
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A109611, A257928, A257938.

Programs

Mathematica

Dv[n_]:=Divisors[Prime[n]+2]
L[n_]:=Length[Dv[n]]
P[k_,n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n],2]]&&Mod[PrimePi[Part[Dv[k*n],2]],n]==0&&PrimeQ[Part[Dv[k*n],3]]&&Mod[PrimePi[Part[Dv[k*n],3]],n]==0
Do[k=0;Label[bb];k=k+1;If[P[k,n],Goto[aa]];Goto[bb];Label[aa];Print[n," ", k];Continue,{n,1,50}]