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A292621 a(n) = a(n-1) + a(floor(log(n))) with a(1) = 1, a(2) = 2.

%I A292621 #18 Sep 28 2017 19:45:35
%S A292621 1,2,3,4,5,6,7,9,11,13,15,17,19,21,23,25,27,29,31,33,36,39,42,45,48,
%T A292621 51,54,57,60,63,66,69,72,75,78,81,84,87,90,93,96,99,102,105,108,111,
%U A292621 114,117,120,123,126,129,132,135,139,143,147,151,155,159,163,167
%N A292621 a(n) = a(n-1) + a(floor(log(n))) with a(1) = 1, a(2) = 2.
%C A292621 a(n) > c*n*log(n)*log(log(n))*log(log(log(n)))*...*log(log...(log(n))...) (k layers) for any sufficient large n, any constant c and any positive integer k.
%C A292621 The sum of 1/a(i) for i = 1, 2, 3, ... diverges extremely slowly.
%H A292621 Robert Israel, <a href="/A292621/b292621.txt">Table of n, a(n) for n = 1..10000</a>
%H A292621 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>
%H A292621 Fedor Petrov, <a href="https://mathoverflow.net/questions/280092">The proof of the divergence of the sum of 1/a(i)</a>, Mathoverflow, Sep 2017.
%p A292621 f:= proc(n) option remember;
%p A292621 procname(n-1)+procname(floor(log(n)))
%p A292621 end proc:
%p A292621 f(1):= 1: f(2):= 2:
%p A292621 map(f, [$1..100]); # _Robert Israel_, Sep 28 2017
%t A292621 a[n_] := a[n] = If[n <= 2, n, a[n - 1] + a[Floor@ Log@ n]]; Array[a, 62] (* _Michael De Vlieger_, Sep 21 2017 *)
%o A292621 (PARI) a(n) = if (n<=2, n, a(n-1) + a(floor(log(n)))); \\ _Michel Marcus_, Sep 21 2017
%Y A292621 Cf. A000195, A031876, A292620.
%K A292621 nonn,easy
%O A292621 1,2
%A A292621 _Yi Yang_, Sep 20 2017