A292621 - OEIS

%I #18 Sep 28 2017 19:45:35

%S 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 51,54,57,60,63,66,69,72,75,78,81,84,87,90,93,96,99,102,105,108,111,

%U 114,117,120,123,126,129,132,135,139,143,147,151,155,159,163,167

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

%C 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 The sum of 1/a(i) for i = 1, 2, 3, ... diverges extremely slowly.

%H Robert Israel, <a href="/A292621/b292621.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>

%H 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 f:= proc(n) option remember;

%p procname(n-1)+procname(floor(log(n)))

%p end proc:

%p f(1):= 1: f(2):= 2:

%p map(f, [$1..100]); # _Robert Israel_, Sep 28 2017

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

%Y Cf. A000195, A031876, A292620.

%K nonn,easy

%O 1,2

%A _Yi Yang_, Sep 20 2017