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 A051918 #10 Dec 10 2014 09:16:44
%S A051918 2,2,2,3,4,5,5,10,13,13,26,32,287,18380,21727,23205,24828,35646,48819,
%T A051918 51476
%N A051918 Start with n, apply k->2k+1 until reach new record prime; sequence gives number of steps needed.
%C A051918 a(20) > 60000. [From _Donovan Johnson_, May 23 2010]
%e A051918 0->1->3, new record prime 3 in 2 steps; 1->3->7, new record prime 7 in 2 steps; 2->5->11, new record prime 11 in 2 steps; 3->7->15->31, new record prime 31 in 3 steps.
%e A051918 For the next few terms we have: 4->9->19->39->79; 5->11->23->47->95->191; 6->13->27->55->111->223; 7->15->31->63->127->255->511->1023->2047->4095->8191; etc.
%t A051918 f[0] = {start=0, k=3, steps=2}; f[n_] := f[n] = (k=start=f[n-1][[1]]+1; steps=0; While[!PrimeQ[k] || k <= f[n-1][[2]], k=2k+1; steps++]; {start, k, steps}); A051918 = Table[Print[f[n] // Last]; f[n], {n, 0, 13}][[All, 3]] (* _Jean-François Alcover_, Dec 10 2014 *)
%Y A051918 Cf. A051919, A052333.
%K A051918 nice,nonn,hard
%O A051918 0,1
%A A051918 _N. J. A. Sloane_, Dec 18 1999
%E A051918 More terms from _Naohiro Nomoto_, May 21 2001
%E A051918 a(13)-a(19) from _Donovan Johnson_, May 23 2010