A013937 - OEIS

%I #32 May 26 2021 09:40:00

%S 0,1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,18,19,20,21,22,23,24,25,27,28,

%T 29,31,32,33,34,35,37,38,39,40,41,42,43,44,46,47,48,49,50,51,52,53,55,

%U 56,57,58,59,60,62,63,65,66,67,68,69,70,71,72,75,76,77,78,79,80,81,82

%N a(n) = Sum_{k=1..n} floor(n/k^3).

%H Seiichi Manyama, <a href="/A013937/b013937.txt">Table of n, a(n) for n = 0..10000</a>

%H Benoit Cloitre, <a href="/A013937/a013937.png">Plot of (a(n)-zeta(3)*n)/n^(1/3)-zeta(1/3)</a>

%F a(n) = a(n-1)+A061704(n). a(n) = Sum_{k=1..n} floor((n/k)^(1/3)) with asymptotic formula: a(n) = zeta(3)*n+zeta(1/3)*n^(1/3)+O(n^theta) where theta<1/3 and we conjecture that theta=1/4+epsilon is the best possible choice. - _Benoit Cloitre_, Nov 05 2012

%F G.f.: (1/(1 - x))*Sum_{k>=1} x^(k^3)/(1 - x^(k^3)). - _Ilya Gutkovskiy_, Feb 11 2017

%e a(36) = [36/1]+[36/8]+[36/27]+[36/64]+... = 36+4+1+0+... = 41.

%p A013937:=n->add(floor(n/k^3), k=1..n); seq(A013937(n), n=0..100); # _Wesley Ivan Hurt_, Feb 15 2014

%t Table[Sum[Floor[n/k^3], {k, n}], {n, 0, 100}] (* _Wesley Ivan Hurt_, Feb 15 2014 *)

%o (PARI) a(n)=sum(k=1,ceil(n^(1/3)),n\k^3) /*_Benoit Cloitre_, Nov 05 2012 */

%Y Cf. A005187, A006218, A011371, A013936, A013939 for similar sequences.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, _Henri Lifchitz_

%E More terms from _Henry Bottomley_, Jul 03 2001