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 1-5 of 5 results.

A075991 Indices where record values of A075989 occur.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 19, 23, 29, 33, 35, 39, 47, 53, 59, 63, 68, 69, 75, 77, 83, 89, 95, 107, 113, 116, 117, 118, 119, 131, 138, 139, 143, 149, 158, 159, 165, 167, 179, 189, 203, 215, 231, 233, 238, 239, 249, 263, 278, 279, 284, 285, 293, 298, 299, 315, 319, 333
Offset: 1

Views

Author

Clark Kimberling, Sep 28 2002

Keywords

Examples

			11 is a term, since A075989(11) = 5 exceeds A075989(h) for h = 1, 2, ..., 10, these values being 0, 0, 1, 0, 2, 1, 2, 2, 3, 2.

Crossrefs

Cf. A075989, A075990.

A271860 a(n) = -Sum_{i=1..n} (-1)^floor(n/i).

Original entry on oeis.org

0, 1, 0, 3, 0, 3, 2, 5, 0, 5, 4, 7, 2, 5, 4, 11, 4, 7, 6, 9, 4, 11, 10, 13, 4, 9, 8, 15, 10, 13, 12, 15, 6, 13, 12, 19, 12, 15, 14, 21, 12, 15, 14, 17, 12, 23, 22, 25, 12, 17, 16, 23, 18, 21, 20, 27, 18, 25, 24, 27, 18, 21, 20, 31, 20, 27, 26, 29, 24, 31, 30
Offset: 0

Views

Author

Wesley Ivan Hurt, Apr 15 2016

Keywords

Links

Crossrefs

Column k=1 of A345033.
Cf. A000035, A075989.

Programs

  • Maple
    A271860:=n->-add((-1)^floor(n/i), i=1..n): seq(A271860(n), n=0..100);
  • Mathematica
    Table[-Sum[(-1)^Floor[n/i], {i, n}], {n, 0, 100}]
  • PARI
    a(n) = -sum(i=1, n, (-1)^(n\i)); \\ Michel Marcus, Apr 16 2016
  • PARI
    my(N=99, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*(1-x^k)/(1+x^k))/(1-x))) \\ Seiichi Manyama, Jun 06 2021
  • Python
    from math import isqrt
def A271860(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)<<1)-n # Chai Wah Wu, Oct 23 2023

Formula

a(2n) = 2*A075989(n) for n>0.
a(n) mod 2 = A000035(n).
a(n) = Sum_{k=1..n} (-1)^floor((n-k)/k). - Wesley Ivan Hurt, May 09 2021
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k * (1 - x^k)/(1 + x^k). - Seiichi Manyama, Jun 06 2021

A075988 Number of integers k satisfying 1 <= k <= n and 0 < frac(n/k) < 1/2, where frac(n/k) is the fractional part of n/k; i.e., frac(n/k) = n/k - floor(n,k).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 3, 4, 4, 4, 7, 5, 5, 8, 9, 6, 10, 8, 10, 12, 10, 10, 14, 13, 13, 13, 15, 13, 19, 16, 16, 18, 16, 17, 24, 20, 18, 20, 23, 21, 25, 23, 23, 25, 25, 23, 30, 26, 28, 30, 28, 26, 30, 30, 34, 34, 32, 28, 37, 35, 31, 36, 37, 37, 41, 35, 37, 37, 41, 38, 46, 42, 40
Offset: 1

Views

Author

Clark Kimberling, Sep 28 2002

Keywords

Crossrefs

Cf. A075989, A076990.

Programs

  • Magma
    [&+[(Ceiling(n/k)-Round(n/k)): k in [1..n]]: n in [1..80]]; // Vincenzo Librandi, Jul 30 2017
  • PARI
    a(n) = sum(k=1, n, f = frac(n/k); f  && (f < 1/2)); \\ Michel Marcus, Jul 30 2017

Formula

Sum_{k=1..n} (ceiling(n/k) - round(n/k)). - Vladeta Jovovic, Mar 01 2004

A339217 a(n) = Sum_{k=1..n} floor((2*n-k)/k).

Original entry on oeis.org

0, 1, 4, 8, 12, 17, 23, 27, 34, 40, 46, 52, 60, 65, 73, 81, 87, 93, 104, 108, 118, 126, 132, 140, 150, 157, 165, 173, 183, 189, 201, 205, 216, 226, 232, 242, 254, 258, 268, 278, 288, 295, 307, 313, 323, 335, 343, 349, 363, 369, 382, 390, 398, 408, 420, 428, 440, 448, 456, 464, 482
Offset: 0

Views

Author

Wesley Ivan Hurt, Dec 22 2020

Keywords

Crossrefs

Cf. A002541, A153485, A279847, A338991, A339370.

Programs

  • Mathematica
    Table[Sum[Floor[(2 n - i)/i], {i, n}], {n, 0, 60}]
  • PARI
    a(n) = sum(k=1, n, (2*n-k)\k); \\ Michel Marcus, Dec 22 2020

Formula

From Vaclav Kotesovec, Dec 23 2020: (Start)
For n>0, a(n) = 2*A006218(n) + A075989(n) - n.
a(n) ~ 2*n * (log(2*n) + 2*gamma - 2), where gamma is the Euler-Mascheroni constant A001620. (End)

A339919 a(n) = Sum_{k=1..n} (floor(3*n/k) - 3*floor(n/k)).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 1, 5, 4, 4, 5, 9, 5, 7, 10, 10, 8, 13, 8, 16, 13, 11, 15, 19, 14, 15, 18, 21, 18, 22, 17, 23, 22, 22, 26, 30, 18, 24, 29, 31, 28, 31, 26, 32, 33, 29, 32, 42, 32, 35, 34, 36, 39, 41, 40, 42, 37, 40, 41, 53, 38, 44, 49, 47, 47, 47, 43, 53, 52
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 23 2020

Keywords

Comments

In general, for m>=1, Sum_{k=1..n} (floor(m*n/k) - m*floor(n/k)) ~ m*n * (log(m) - H(m) + 1), where H(m) = A001008(m)/A002805(m) is the m-th harmonic number.

Links

Crossrefs

Cf. A006218, A059851, A075989, A339918.

Programs

  • Mathematica
    Table[Sum[Floor[3*n/k] - 3*Floor[n/k], {k, 1, n}], {n, 0, 100}]
  • PARI
    a(n) = sum(k=1, n, floor(3*n/k) - 3*floor(n/k)); \\ Michel Marcus, Dec 23 2020

Formula

a(n) ~ (3*log(3) - 5/2) * n.
Showing 1-5 of 5 results.

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