This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A260753 #20 Aug 18 2015 11:14:59
%S A260753 2,2,2279,5806,4,1135,816,6556,725,2,1333,10839,27,829,2279,2838,3881,
%T A260753 6540,2564,2,7830,6540,27,4905,6121,8220,316,1061,2,14691,2,1168,738,
%U A260753 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 A260753 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 A260753 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 A260753 In the case s = -1 and t = 1, the conjecture implies that A261136 has infinitely many terms.
%D A260753 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 A260753 Zhi-Wei Sun, <a href="/A260753/b260753.txt">Table of n, a(n) for n = 1..5000</a>
%H A260753 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 A260753 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 A260753 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 A260753 f[n_]:=Prime[Prime[n]]-Prime[n]+1
%t A260753 PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
%t A260753 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 A260753 Cf. A000040, A234694, A234695, A236832, A238766, A238878, A261136, A261362.
%K A260753 nonn
%O A260753 1,1
%A A260753 _Zhi-Wei Sun_, Aug 18 2015