A260753 - OEIS

%I #20 Aug 18 2015 11:14:59

%S 2,2,2279,5806,4,1135,816,6556,725,2,1333,10839,27,829,2279,2838,3881,

%T 6540,2564,2,7830,6540,27,4905,6121,8220,316,1061,2,14691,2,1168,738,

%U 4707,785,12467,5492,1447,542,538,12840,829,4732,5637,785,1246,1198,433,58,573,26280,17387,316,430,1198,4315,4315,1479,4315,1497

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

%C Conjecture: For any s and t in the set {1,-1}, every positive rational number r can be written as m/n with m and n in the set {k>0: prime(prime(k))+s*prime(k)+t = prime(p) for some prime p}.

%C In the case s = -1 and t = 1, the conjecture implies that A261136 has infinitely many terms.

%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="/A260753/b260753.txt">Table of n, a(n) for n = 1..5000</a>

%H Zhi-Wei Sun, <a href="/A260753/a260753.txt">Checking the conjecture for s = -1, t = 1 and r = a/b (a,b = 1..125)</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) = 2279 since  prime(prime(2279))-prime(2279)+1 = prime(20147)-20147+1 = 226553-20146 = 206407 = prime(18503) with 18503 prime, and  prime(prime(2279*3))-prime(2279*3)+1 = prime(68777)-68777+1 = 865757-68776 = 796981 = prime(63737) with 63737 prime.

%t f[n_]:=Prime[Prime[n]]-Prime[n]+1

%t PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]

%t 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,1,60}]

%Y Cf. A000040, A234694, A234695, A236832, A238766, A238878, A261136, A261362.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Aug 18 2015