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 A227615 #18 Jun 27 2021 04:42:16
%S A227615 3,4,8,16,31,61,122,244,487,973,1946,3892,7783,15565,31130,62259,
%T A227615 124517,249033,498066,996131,1992262,3984524,7969047,15938093,
%U A227615 31876185,63752369,127504737,255009473,510018945,1020037890,2040075780
%N A227615 Number of bits necessary to represent u(n) in binary, where u is the Lucas-Lehmer sequence: u(0) = 100 (in binary); for n>0, u(n) = u(n-1)^2 - 2.
%C A227615 a(0)=3, a(1)=4 and for n>=1, a(n+1) is 2*a(n) or 2*a(n)-1.
%C A227615 It seems the rule to decide between the 2 is not straightforward. So you actually need to compute u(n) to have its required number of bits.
%C A227615 Yet, for n>=1, we have a lower bound: a(n) >= 2^n and an upper bound: a(n) <= 2^(n+1).
%F A227615 a(n) = E(log_2(u(n))) + 1, where E(x) is the integer part of x and u is defined by: u(0) = 4 (or 100 in binary) and for n>0, u(n) = u(n-1)^2 - 2.
%F A227615 a(n) = A070939(A003010(n)). - _Michel Marcus_, Apr 04 2016
%e A227615 For n=2, u(2) = 194, log_2(u(2)) is between 7.5 and 7.6, so E(log_2(u(2))) = 7, so a(2) = E(log_2(u(2))) + 1 = 8. And indeed, u(2) = 194 (in base 10) = 11000010 in base 2 requires 8 bits (all bits above are 0).
%o A227615 (PARI) lista(nn) = {a = 4; print1(#binary(a), ", "); for (n=1, nn, a = a^2-2; print1(#binary(a), ", "););} \\ _Michel Marcus_, Apr 04 2016
%Y A227615 Cf. A003010, A070939, A227608.
%K A227615 nonn,base
%O A227615 0,1
%A A227615 _Olivier de Mouzon_, Jul 17 2013
%E A227615 Terms from a(19) on from _Michel Marcus_, Apr 04 2016