A261295 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m) = prime(p)+2 for some prime p}.

3, 3, 6, 578, 18, 3, 6, 90, 1868, 374, 4, 674, 278, 3, 6, 114, 3534, 110, 6, 354, 4, 14, 28464, 2790, 84, 4452, 2802, 3, 6, 3, 90, 2820, 354, 110, 4080, 278, 44, 3, 2712, 18, 3012, 90, 14, 12672, 44, 14, 1572, 1124, 720, 42, 114, 44, 84, 2790, 42, 90, 42, 3, 6, 84, 44, 1572, 3068, 1742, 2394, 174, 110, 744, 3020, 578

Offset: 1

Author

Zhi-Wei Sun, Aug 14 2015

Keywords

• nonn

Comments

Conjecture: (i) Any positive rational number r can be written as m/n with m and n in the set S = {k>0: prime(k) = prime(p)+2 for some prime p} = {p+1: p and prime(p)+2 are both prime}.

(ii) Any positive rational number r can be written as m/n with m and n in the set T = {k>0: prime(k) = prime(p)-2 for some prime p} = {p-1: p and prime(p)-2 are both prime}.

(iii) Any positive rational number r not equal to 1 can be written as m/n with m in S and n in T, where the sets S and T are given in parts (i) and (ii).

For example, 4/5 = 15648/19560 with 15647, prime(15647)+2 = 171763, 19559 and prime(19559)+2 = 219409 all prime; and 4/5 = 67536/84420 with 67537, prime(67537)-2 = 848849, 84421 and prime(84421)-2 = 1081937 all prime. Also, 4/5 = 8/10 with 7, prime(7)+2 = 19, 11 and prime(11)-2 = 29 all prime; and 5/4 = 8220/6576 with 8221, prime(8221)+2 = 84349, 6577 and prime(6577)-2 = 65837 all prime.

Examples

			a(3) = 6 since prime(6) = 13 = prime(5)+2 with 5 prime, and prime(6*3) = 61 = prime(17)+2 with 17 prime.
a(4) = 578 since prime(578) = 4219 = prime(577)+2 with 577 prime, and prime(578*4) = 20479 = prime(2311)+2 with 2311 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..2000
• Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..200
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A001359, A006512, A218829, A258803, A259487, A259540, A261281, A261282.

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]-2]&&PQ[f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]