A292620 - OEIS

%I #19 Sep 28 2017 19:45:27

%S 1,2,3,5,7,9,11,14,17,20,23,26,29,32,35,40,45,50,55,60,65,70,75,80,85,

%T 90,95,100,105,110,115,122,129,136,143,150,157,164,171,178,185,192,

%U 199,206,213,220,227,234,241,248,255,262,269,276,283,290,297,304,311

%N a(n) = a(n-1) + a(floor(log_2(n))), with a(1) = 1.

%C a(n) > c*n*log_2(n)*log_2(log_2(n))*log_2(log_2(log_2(n)))*...*log_2(log_2...(log_2(n))...) (k layers) for any sufficiently large n, any constant c and any positive integer k.

%C The sum of 1/a(i) for i = 1, 2, 3, ... converges extremely slowly to the limit 5.70....

%H Robert Israel, <a href="/A292620/b292620.txt">Table of n, a(n) for n = 1..10000</a>

%H KeyTo9_Fans, <a href="http://bbs.emath.ac.cn/thread-9628-1-1.html">A Chinese post discussing the sum of 1/a(i)</a>

%p f:= proc(n) option remember; procname(n-1)+procname(ilog2(n)) end proc:

%p f(1):= 1:

%p map(f, [$1..100]); # _Robert Israel_, Sep 24 2017

%t a[n_] := a[n] = If[n == 1, 1, a[n - 1] + a[Floor@ Log2@ n]]; Array[a, 59] (* _Michael De Vlieger_, Sep 21 2017 *)

%o (PARI) a(n) = if (n<=2, n, a(n-1) + a(logint(n, 2))); \\ _Michel Marcus_, Sep 21 2017

%Y Cf. A000523, A292621.

%K nonn,easy

%O 1,2

%A _Yi Yang_, Sep 20 2017