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 A258836 #13 Jun 15 2015 06:59:52
%S A258836 4,6,4,18,6,12,6,30,12,6,18,6,150,30,4,12,60,4,12,12,42,30,240,18,6,
%T A258836 12,4,270,12,6,42,6,6,30,12,12,180,6,60,6,30,150,30,30,4,18,2550,4,18,
%U A258836 12,42,6,150,30,12,60,4,6,60,4,462,180,1230,18,30,108,60,180,12,6,30,6,570,420,462,180,6,4,198,42,522,600,1050,42,12,12,4,60,432,18,12,60,30,60,6,12,150,60,30,6
%N A258836 Least practical number q with q-1 and q+1 twin prime such that n = q'/q for some practical number q' with q'-1 and q'+1 twin prime.
%C A258836 Conjecture: a(n) exists for any n > 0. Moreover, any positive rational number r can be written as q'/q, where q and q' are terms of A258838 (i.e., q is practical with q-1 and q+1 twin prime, and q' is practical with q'-1 and q'+1 twin prime).
%C A258836 This implies that there are infinitely many "sandwiches of the second kind" (i.e., triples {q-1,q,q+1} with q practical and q-1 and q+1 twin prime).
%C A258836 I have verified the conjecture for all those rational numbers r = n/m with m,n = 1,...,1000. -_Zhi-Wei Sun_, Jun 15 2015
%H A258836 Zhi-Wei Sun, <a href="/A258836/b258836.txt">Table of n, a(n) for n = 1..10000</a>
%H A258836 Zhi-Wei Sun, <a href="/A258836/a258836.txt">Checking the conjecture for r = n/m with 1 <= n <= m <= 1000</a>
%H A258836 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;20e70044.1301">Sandwiches with primes and practical numbers</a>, a message to Number Theory List, Jan. 13, 2013.
%H A258836 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A258836 a(1) = 4 since 1 = 4/4 with 4 practical and 4-1 and 4+1 twin prime.
%e A258836 a(2) = 6 since 2 = 12/6, 6 is practical with 6-1 and 6+1 twin prime, and 12 is practical with 12-1 and 12+1 twin prime.
%t A258836 f[n_]:=FactorInteger[n]
%t A258836 Pow[n_,i_]:=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t A258836 Con[n_]:=Sum[If[Part[Part[f[n],s+1],1]<=DivisorSigma[1,Product[Pow[n,i],{i,1,s}]]+1,0,1],{s,1,Length[f[n]]-1}]
%t A258836 pr[n_]:=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t A258836 SW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&pr[n]
%t A258836 Do[k=0;Label[bb];k=k+1;If[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&SW[n*(Prime[k]+1)],Goto[aa],Goto[bb]];
%t A258836 Label[aa];Print[n," ",Prime[k]+1];Continue,{n,1,100}]
%Y A258836 Cf. A000040, A001359, A006512, A005153, A210479, A258803, A258811, A258838.
%K A258836 nonn
%O A258836 1,1
%A A258836 _Zhi-Wei Sun_, Jun 11 2015