cp's OEIS Frontend

A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).

%I A069470 #21 Sep 08 2022 08:45:05
%S A069470 0,1,2,4,5,6,9,10,11,13,15,16,19,20,21,24,25,26,29,30,32,35,36,37,40,
%T A069470 41,42,44,46,47,52,53,54,56,57,58,62,63,64,66,68,69,73,74,75,79,80,81,
%U A069470 84,85,87,89,90,91,94,96,98,100,101,102,107,108,109,112,113,114,118
%N A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).
%C A069470 The summation has floor(1/2 + sqrt(2*n)) = A002024(n) nonzero terms. - _Enrique Pérez Herrero_, Apr 05 2010
%H A069470 G. C. Greubel, <a href="/A069470/b069470.txt">Table of n, a(n) for n = 0..10000</a>
%F A069470 a(n) = a(n-1) + A007862(n).
%F A069470 It appears that limit((sum(floor((1/2)*n/(k*(k+1))), k=1..n))/n, n=infinity) = 1/2. - _Stephen Crowley_, Aug 12 2009
%F A069470 From _Enrique Pérez Herrero_, Apr 05 2010: (Start)
%F A069470 a(n) <= floor((2*n^2)/(1 + n)) = A004275(n).
%F A069470 a(n) <= floor((2*n*floor((1 + 2*sqrt(2*n))/2))/(1+floor((1+2*sqrt(2*n))/2))). (End)
%F A069470 G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^(k*(k+1)/2)). - _Ilya Gutkovskiy_, Jul 11 2019
%e A069470 a(11) = floor(11/1) + floor(11/3) + floor(11/6) + floor(11/10) + floor(11/15) + ... = 11 + 3 + 1 + 1 + 0 + ... = 16.
%t A069470 A069470[n_]:=Sum[Floor[(2*n)/(k*(1 + k))], {k, 1, Floor[1/2 + Sqrt[2*n]]}] (* _Enrique Pérez Herrero_, Apr 05 2010 *)
%o A069470 (PARI) for(n=0, 30, print1(sum(k=1, 100, floor(n/(k*(k+1)/2))), ", ")) \\ _G. C. Greubel_, May 23 2018
%o A069470 (Magma) [(&+[Floor(n/(k*(k+1)/2)): k in [1..100]]): n in [0..30]]; // _G. C. Greubel_, May 23 2018
%Y A069470 Cf. A000217, A006218, A007862, A013936, A013937, A013938, A013939.
%Y A069470 Cf. A002024, A004275. - _Enrique Pérez Herrero_, Apr 05 2010
%K A069470 nonn
%O A069470 0,3
%A A069470 _Henry Bottomley_, Mar 25 2002