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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}.
1

%I #14 Aug 14 2015 14:05:10

%S 3,3,6,578,18,3,6,90,1868,374,4,674,278,3,6,114,3534,110,6,354,4,14,

%T 28464,2790,84,4452,2802,3,6,3,90,2820,354,110,4080,278,44,3,2712,18,

%U 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

%N 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}.

%C 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}.

%C (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}.

%C (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).

%C 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.

%D 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.

%H Zhi-Wei Sun, <a href="/A261295/b261295.txt">Table of n, a(n) for n = 1..2000</a>

%H Zhi-Wei Sun, <a href="/A261295/a261295_1.txt">Checking the conjecture for r = a/b with a,b = 1..200</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

%e a(3) = 6 since prime(6) = 13 = prime(5)+2 with 5 prime, and prime(6*3) = 61 = prime(17)+2 with 17 prime.

%e a(4) = 578 since prime(578) = 4219 = prime(577)+2 with 577 prime, and prime(578*4) = 20479 = prime(2311)+2 with 2311 prime.

%t f[n_]:=Prime[n]

%t PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]

%t 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}]

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

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Aug 14 2015