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A059851
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a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + ... (this is a finite sum).
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25
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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
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OFFSET
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0,4
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COMMENTS
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As n goes to infinity we have the asymptotic formula: a(n) ~ n * log(2).
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LINKS
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FORMULA
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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
a(n) = Sum_{k=1..n} (floor(n/k) mod 2)
a(n) = Sum_{k=1..n} round(n/(2*k)) - floor(n/(2*k)), where round(1/2)=1. (End)
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EXAMPLE
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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.
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MAPLE
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for n from 0 to 200 do printf(`%d, `, sum((-1)^(i+1)*floor(n/i), i=1..n)) od:
{ 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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001
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EXTENSIONS
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More terms from James A. Sellers and Larry Reeves (larryr(AT)acm.org), Feb 27 2001
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STATUS
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approved
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