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 A051919 #8 Dec 10 2014 09:17:02
%S A051919 3,7,11,31,79,191,223,8191,73727,81919,738197503,51539607551,
%T A051919 3232601036663613174009984612954335460652964980182760163363015849805536730581466940964863
%N A051919 Start with n, apply k->2k+1 until reach new record prime; sequence gives record primes.
%e A051919 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; ...
%t A051919 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}); A051919 = Table[Print[f[n] // Last]; f[n], {n, 0, 12}][[All, 2]] (* _Jean-François Alcover_, Dec 10 2014 *)
%Y A051919 Cf. A051918, A052333.
%K A051919 nice,nonn
%O A051919 0,1
%A A051919 _N. J. A. Sloane_, Dec 18 1999
%E A051919 More terms from _Naohiro Nomoto_, May 21 2001, who reports that the next term is > 10^130.