A261362 Least positive integer k such that both k and k*n belong to the set {m>0: 2*prime(prime(m))+1 = prime(p) for some prime p}.

2, 21531, 2, 35434, 11107, 35175, 24674, 64624, 127943, 1981, 155709, 50657, 74313, 11479, 6, 1981, 43405, 40859, 74229, 2, 154292, 51711, 29460, 29011, 42001, 28352, 2979, 85836, 6936, 186608, 3705, 14402, 25525, 96192, 6, 113433, 164, 787, 71873, 3365, 93169, 47219, 43128, 184740, 2, 78329, 13656, 6936, 139469, 26713

Offset: 1

Author

Zhi-Wei Sun, Aug 16 2015

Keywords

• nonn

Comments

Conjecture: Let a,b,c be positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. If a+b+c is even and a is not equal to b, then any positive rational number r can be written as m/n with m and n in the set {k>0: a*prime(p) - b*prime(prime(k)) = c for some prime p}.

This implies the conjecture in A261361.

Examples

			a(2) = 21531 since 2*prime(prime(21531))+1 = 2*prime(243799)+1 = 2*3403703+1 = 6807407 = prime(464351) with 464351 prime, and 2*prime(prime(21531*2))+1 = 2*prime(520019)+1 = 2*7686083+1 = 15372167 = prime(993197) with 993197 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..2250
• Zhi-Wei Sun, Checking the conjecture for (a,b,c) = (1,2,1) and r = s/t (s,t = 1..70)
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A005384, A260886, A261353, A261354, A261361.

Programs

Mathematica

f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]]&&PQ[f[k*n]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,50}]