A075997 a(n) = [n/2] - [n/3] + [n/4] - [n/5] + [n/6] - ..., where [n/k] = floor(n/k).

0, 0, 1, 0, 2, 1, 2, 1, 4, 2, 3, 2, 5, 4, 5, 2, 6, 5, 6, 5, 8, 5, 6, 5, 10, 8, 9, 6, 9, 8, 9, 8, 13, 10, 11, 8, 12, 11, 12, 9, 14, 13, 14, 13, 16, 11, 12, 11, 18, 16, 17, 14, 17, 16, 17, 14, 19, 16, 17, 16, 21, 20, 21, 16, 22, 19, 20, 19, 22, 19, 20, 19, 26, 25, 26, 21, 24, 21, 22, 21

Offset: 0

Author

Clark Kimberling, Sep 28 2002

Keywords

• nonn
• easy

Comments

a(n) is the number of terms among {floor(n/k)}, 1<=k<=n, which are even. - Leroy Quet, Jan 19 2006

Examples

			a(6) = [6/2]-[6/3]+[6/4]-[6/5]+[6/6]-[6/7]+... = 3-2+1-1+1-0+... = 2.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000005, A006218, A060831, A059851.

Programs

Maple

A075997:=n->add(floor(n/(2*i))-floor((n-i)/(2*i)), i=1..n): seq(A075997(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2016

Mathematica

Table[Sum[Floor[n/(2 i)] - Floor[(n - i)/(2 i)], {i, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2016 *)

PARI

a(n) = sum(k=2, n, (-1)^k*(n\k)); \\ Michel Marcus, Dec 20 2020

Python

from math import isqrt
def A075997(n): return n+(s:=isqrt(n))**2-((t:=isqrt(m:=n>>1))**2<<1)-(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

Formula

a(n) = n - A059851(n).

a(n) = n - A006218(n) + 2*A006218(floor(n/2)). - Vladeta Jovovic, Oct 02 2002

a(n) = n - Sum_{n/2A000005(k). - Leroy Quet, Jan 19 2006

G.f.: ( Sum_{i>0} x^(2*i)/(1+x^i) )/(1-x). - Vladeta Jovovic, Apr 24 2006

a(n) = Sum_{i=1..n} floor(n/(2*i)) - floor((n-i)/(2*i)). - Wesley Ivan Hurt, Jan 30 2016

Conjecture: Let f(a,b)=1, if (a+b) mod |a-b| != (a mod |a-b|)+(b mod |a-b|), and 0 otherwise. a(n) = Sum_{k=1..n-1} f(n+k,n-k). - Benedict W. J. Irwin, Sep 23 2016

a(n) = Sum_{k=1..n} (floor((n-i)/i) mod 2 ). - Wesley Ivan Hurt, Dec 20 2020

a(n) ~ (1 - log(2))*n. - Vaclav Kotesovec, Jun 14 2025