cp's OEIS Frontend

A059851 a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + ... (this is a finite sum).

%I A059851 #55 Jul 02 2025 16:02:01
%S A059851 0,1,1,3,2,4,4,6,4,7,7,9,7,9,9,13,10,12,12,14,12,16,16,18,14,17,17,21,
%T A059851 19,21,21,23,19,23,23,27,24,26,26,30,26,28,28,30,28,34,34,36,30,33,33,
%U A059851 37,35,37,37,41,37,41,41,43,39,41,41,47,42,46,46,48,46,50,50,52,46,48,48
%N A059851 a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + ... (this is a finite sum).
%C A059851 As n goes to infinity we have the asymptotic formula: a(n) ~ n * log(2).
%C A059851 More precisely, a(n) = n * log(2) + O(n^(131/416) * (log n)^(26947/8320)). - _V Sai Prabhav_, Jun 02 2025
%H A059851 T. D. Noe, <a href="/A059851/b059851.txt">Table of n, a(n) for n = 0..10000</a>
%H A059851 V Sai Prabhav, <a href="/A059851/a059851.pdf">Asymptotic Expansion of a(n)</a>
%F A059851 From _Vladeta Jovovic_, Oct 15 2002: (Start)
%F A059851 a(n) = A006218(n) - 2*A006218(floor(n/2)).
%F A059851 G.f.: 1/(1-x)*Sum_{n>=1} x^n/(1+x^n). (End)
%F A059851 a(n) = Sum_{n/2 < k < =n} d(k) - Sum_{1 < =k <= n/2} d(k), where d(k) = A000005(k). Also, a(n) = number of terms among {floor(n/k)}, 1<=k<=n, that are odd. - _Leroy Quet_, Jan 19 2006
%F A059851 From _Ridouane Oudra_, Aug 15 2019: (Start)
%F A059851 a(n) = Sum_{k=1..n} (floor(n/k) mod 2).
%F A059851 a(n) = (1/2)*(n + A271860(n)).
%F A059851 a(n) =  Sum_{k=1..n} round(n/(2*k)) - floor(n/(2*k)), where round(1/2) = 1. (End)
%F A059851 a(n) = 2*A263086(n) - 3*A006218(n). - _Ridouane Oudra_, Aug 17 2024
%e A059851 a(5) = 4 because floor(5) - floor(5/2) + floor(5/3) - floor(5/4) + floor(5/5) - floor(5/6) + ... = 5 - 2 + 1 - 1 + 1 - 0 + 0 - 0 + ... = 4.
%p A059851 for n from 0 to 200 do printf(`%d,`, sum((-1)^(i+1)*floor(n/i), i=1..n)) od:
%t A059851 f[list_, i_] := list[[i]]; nn = 200; a = Table[1, {n, 1, nn}]; b =
%t A059851 Table[If[OddQ[n], 1, -1], {n, 1, nn}];Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] // Accumulate (* _Geoffrey Critzer_, Mar 29 2015 *)
%t A059851 Table[Sum[Floor[n/k] - 2*Floor[n/(2*k)], {k, 1, n}], {n, 0, 100}] (* _Vaclav Kotesovec_, Dec 23 2020 *)
%o A059851 (PARI) { for (n=0, 10000, s=1; d=2; a=n; while ((f=floor(n/d)) > 0, a-=s*f; s=-s; d++); write("b059851.txt", n, " ", a); ) } \\ _Harry J. Smith_, Jun 29 2009
%o A059851 (Python)
%o A059851 from math import isqrt
%o A059851 def A059851(n): return ((t:=isqrt(m:=n>>1))**2<<1)-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # _Chai Wah Wu_, Oct 23 2023
%o A059851 (Magma)
%o A059851 A059851:= func< n | (&+[Floor(n/j)*(-1)^(j-1): j in [1..n]]) >;
%o A059851 [A059851(n): n in [1..80]]; // _G. C. Greubel_, Jun 27 2024
%o A059851 (SageMath)
%o A059851 def A059851(n): return sum((n//j)*(-1)^(j-1) for j in range(1,n+1))
%o A059851 [A059851(n) for n in range(81)] # _G. C. Greubel_, Jun 27 2024
%Y A059851 Cf. A000005, A075997, A271860, A263086.
%Y A059851 Partial sums of A048272.
%Y A059851 Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), this sequence (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).
%K A059851 nonn,easy
%O A059851 0,4
%A A059851 Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001
%E A059851 More terms from _James Sellers_ and Larry Reeves (larryr(AT)acm.org), Feb 27 2001