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

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, 19, 21, 21, 23, 19, 23, 23, 27, 24, 26, 26, 30, 26, 28, 28, 30, 28, 34, 34, 36, 30, 33, 33, 37, 35, 37, 37, 41, 37, 41, 41, 43, 39, 41, 41, 47, 42, 46, 46, 48, 46, 50, 50, 52, 46, 48, 48

Offset

0,4

Comments

As n goes to infinity we have the asymptotic formula: a(n) ~ n * log(2).

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Formula

From Vladeta Jovovic, Oct 15 2002: (Start)

a(n) = A006218(n) - 2*A006218(floor(n/2)).

G.f.: 1/(1-x)*Sum_{n>=1} x^n/(1+x^n). (End)

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

From Ridouane Oudra, Aug 15 2019: (Start)

a(n) = Sum_{k=1..n} (floor(n/k) mod 2).

a(n) = (1/2)*(n + A271860(n)).

a(n) = Sum_{k=1..n} round(n/(2*k)) - floor(n/(2*k)), where round(1/2) = 1. (End)

a(n) = 2*A263086(n) - 3*A006218(n). - Ridouane Oudra, Aug 17 2024

Example

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.

Maple

for n from 0 to 200 do printf(`%d, `, sum((-1)^(i+1)*floor(n/i), i=1..n)) od:

Mathematica

f[list_, i_] := list[[i]]; nn = 200; a = Table[1, {n, 1, nn}]; b =
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 *)
Table[Sum[Floor[n/k] - 2*Floor[n/(2*k)], {k, 1, n}], {n, 0, 100}] (* Vaclav Kotesovec, Dec 23 2020 *)

Prog

(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
(Python)
from math import isqrt
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
(Magma)
A059851:= func< n | (&+[Floor(n/j)*(-1)^(j-1): j in [1..n]]) >;
[A059851(n): n in [1..80]]; // G. C. Greubel, Jun 27 2024
(SageMath)
def A059851(n): return sum((n//j)*(-1)^(j-1) for j in range(1, n+1))
[A059851(n) for n in range(81)] # G. C. Greubel, Jun 27 2024

Crossrefs

Cf. A000005, A075997, A271860, A263086.

Partial sums of A048272.

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).

Sequence in context: A069745 A112199 A145815 * A327637 A366409 A345082

Adjacent sequences: A059848 A059849 A059850 * A059852 A059853 A059854

Keyword

nonn,easy

Author

Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

Extensions

More terms from James A. Sellers and Larry Reeves (larryr(AT)acm.org), Feb 27 2001

Status

approved