cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

As n goes to infinity we have the asymptotic formula: a(n) ~ n * log(2).
More precisely, a(n) = n * log(2) + O(n^(131/416) * (log n)^(26947/8320)). - V Sai Prabhav, Jun 02 2025

Examples

			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.

Links

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).

Programs

  • 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
  • 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 *)
  • 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
  • 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

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

Extensions

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

A325939 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k).

Original entry on oeis.org

0, 1, -1, 2, -1, 1, -1, 3, -2, 1, -1, 3, -1, 1, -3, 4, -1, 1, -1, 3, -3, 1, -1, 5, -2, 1, -3, 3, -1, 1, -1, 5, -3, 1, -3, 4, -1, 1, -3, 5, -1, 1, -1, 3, -5, 1, -1, 7, -2, 1, -3, 3, -1, 1, -3, 5, -3, 1, -1, 5, -1, 1, -5, 6, -3, 1, -1, 3, -3, 1, -1, 7, -1, 1, -5, 3, -3, 1, -1, 7
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 09 2019

Keywords

Comments

Number of even divisors of n minus number of odd strong divisors of n (i.e. odd divisors > 1).

Links

Crossrefs

Cf. A032741, A048272, A075997 (partial sums), A325937.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^# &, # > 1 &], {n, 1, 80}]
  • PARI
    A325939(n) = sumdiv(n, d, if(1==d,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019

Formula

G.f.: Sum_{k>=2} (-1)^k * x^k / (1 - x^k).
a(n) = Sum_{d|n, d>1} (-1)^d.
a(n) = 1 - A048272(n).

A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 2, 5, 3, 4, 3, 6, 5, 6, 3, 7, 6, 7, 6, 9, 6, 7, 6, 11, 9, 10, 7, 10, 9, 10, 9, 14, 11, 12, 9, 13, 12, 13, 10, 15, 14, 15, 14, 17, 12, 13, 12, 19, 17, 18, 15, 18, 17, 18, 15, 20, 17, 18, 17, 22, 21, 22, 17, 23, 20, 21, 20, 23, 20, 21, 20, 27, 26, 27, 22, 25, 22, 23, 22
Offset: 1

Views

Author

Ilya Gutkovskiy, May 25 2020

Keywords

Comments

a(n) = number of terms among {ceiling(n/k)}, 1 <= k <= n, that are odd.

Links

Crossrefs

Cf. A002541, A006218, A048272, A059851, A075997, A120885, A320226, A325939, A332682, A333194.

Programs

  • Maple
    b:= n-> add((-1)^d, d=numtheory[divisors](n)):
a:= proc(n) option remember; `if`(n>0, 1+b(n-1)+a(n-1), 0) end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 25 2020
  • Mathematica
    Table[Sum[Mod[Ceiling[n/k], 2], {k, 1, n}], {n, 1, 80}]
Table[n - Sum[DivisorSum[k, (-1)^(# + 1) &], {k, 1, n - 1}], {n, 1, 80}]
nmax = 80; CoefficientList[Series[x/(1 - x) (1 + Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sum(k=1, n, ceil(n/k) % 2); \\ Michel Marcus, May 25 2020
  • Python
    from math import isqrt
def A330926(n): return n+(s:=isqrt(n-1))**2-((t:=isqrt(m:=n-1>>1))**2<<1)-(sum((n-1)//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

G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} x^(2*k) / (1 + x^k)).
a(n) = n - Sum_{k=1..n-1} A048272(k).
a(n) = A075997(n-1) + 1.
Showing 1-3 of 3 results.

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