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A063511 a(n) = a(floor(sqrt(n))) * 2.

%I A063511 #31 Nov 30 2024 06:00:35
%S A063511 1,2,2,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%T A063511 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%U A063511 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8
%N A063511 a(n) = a(floor(sqrt(n))) * 2.
%C A063511 From _Kevin Ryde_, May 11 2020: (Start)
%C A063511 The sqrt steps in the definition are equivalent to A211667 but here factors of 2 instead of counting, so a(n) = 2^A211667(n).  A211667 is a double logarithm and the effect of power 2^ is to turn the second into a rounding.  So a(n) is the bit length of n (see A070939) increased to the next power of 2 if not already a power of 2.  Each n = 2^(2^k) is a new high a(n) = 2^(k+1), since such an n is bit length 2^k+1.
%C A063511 In a microcomputer, it's common for machine words to be power-of-2 sizes such as 16, 32, 64, 128 bits.  a(n) can be thought of as the word size needed to contain integer n.  Some algorithms by their nature expect power-of-2 sizes, for example Schönhage and Strassen's big integer multiplication.
%C A063511 This sequence differs from A334789 (2^log*(n)) for n>=256.  For example a(256)=16 whereas A334789(256)=8.  The respective exponent sequences are A211667 (for here) and A001069 (for A334789) which likewise differ for n>=256.
%C A063511 (End)
%H A063511 Kevin Ryde, <a href="/A063511/b063511.txt">Table of n, a(n) for n = 1..8192</a>
%H A063511 Martin Fürer, <a href="http://web.archive.org/web/1id_/http://www.cse.psu.edu/~furer/Papers/mult.pdf">Faster integer multiplication</a>, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, June 11-13 2007.  And <a href="http://dx.doi.org/10.1137/070711761">in SIAM Journal of Computing</a>, volume 30, number 3, 2009, pages 979-1005.  (See size "n" calculation at the start of Algorithm Integer-Multiplication.)
%H A063511 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A063511 a(n) = 2^A211667(n) = 2^ceiling(log_2(log_2(n+1))). - _Kevin Ryde_, May 11 2020
%F A063511 log_2(n+1) <= a(n) < 2*log_2(n+1). - _Charles R Greathouse IV_, Nov 30 2024
%o A063511 (PARI) a(n) = if(n==1,1, 2<<logint(logint(n,2),2)); \\ _Kevin Ryde_, May 11 2020
%Y A063511 Cf. A001146 (indices of new highs), A334789.
%K A063511 easy,nonn
%O A063511 1,2
%A A063511 _Reinhard Zumkeller_, Jul 30 2001
%E A063511 Formula and code by Charles R Greathouse IV moved to A334789 where they apply. - _Kevin Ryde_, May 11 2020