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 1-4 of 4 results.

A055682 a(n) = floor(n*sqrt(n)) - sigma(n), where sigma(n) is the sum of the divisors of n (A000203).

Original entry on oeis.org

0, -1, 1, 1, 5, 2, 10, 7, 14, 13, 24, 13, 32, 28, 34, 33, 52, 37, 62, 47, 64, 67, 86, 57, 94, 90, 100, 92, 126, 92, 140, 118, 141, 144, 159, 125, 187, 174, 187, 162, 220, 176, 237, 207, 223, 239, 274, 208, 286, 260, 292, 276, 331, 276, 335, 299
Offset: 1

Views

Author

N. J. A. Sloane, Nov 29 2000

Keywords

Comments

Always > 0 for n > 2.

References

  • József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 77, section III.1.1.b.

Links

Crossrefs

Cf. A000203, A057641, A058208.

Programs

  • Mathematica
    a[n_] :=  Floor[n*Sqrt[n]] - DivisorSigma[1, n]; Array[a, 100] (* Amiram Eldar, Apr 25 2024 *)
  • PARI
    a(n)=sqrtint(n^3)-sigma(n) \\ Charles R Greathouse IV, Feb 14 2013

A079528 a(n) = sigma(n) - ceiling(n + sqrt n).

Original entry on oeis.org

-1, -1, -1, 1, -2, 3, -2, 4, 1, 4, -3, 12, -3, 6, 5, 11, -4, 16, -4, 17, 6, 9, -4, 31, 1, 10, 7, 22, -5, 36, -5, 25, 9, 14, 7, 49, -6, 15, 10, 43, -6, 47, -6, 33, 26, 19, -6, 69, 1, 35, 13, 38, -7, 58, 9, 56, 15, 24, -7, 100, -7, 26, 33, 55, 10, 69, -8, 49, 18, 65, -8, 114, -8, 31, 40, 55, 10, 81, -8, 97
Offset: 1

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Comments

a(n) >= 0 if n composite.

References

  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.1.1.a.
  • W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964.

Links

Crossrefs

Cf. A079529, A055682, A058208.

Programs

  • Magma
    [SumOfDivisors(n)- Ceiling(n + Sqrt (n)): n in [1..100]]; // Vincenzo Librandi, Dec 13 2014
  • Mathematica
    Table[DivisorSigma[1, n] -Ceiling[n +Sqrt[n]], {n, 1, 80}] (* G. C. Greubel, Jan 15 2019 *)
  • PARI
    vector(80, n, sigma(n) - ceil(n + sqrt(n))) \\ Michel Marcus, Dec 12 2014
  • Sage
    [sigma(n,1) - ceil(n+sqrt(n)) for n in (1..80)] # G. C. Greubel, Jan 15 2019

A079529 a(n) = sigma(n) - ceiling(n + sqrt n) as n runs through the composite numbers A002808.

Original entry on oeis.org

1, 3, 4, 1, 4, 12, 6, 5, 11, 16, 17, 6, 9, 31, 1, 10, 7, 22, 36, 25, 9, 14, 7, 49, 15, 10, 43, 47, 33, 26, 19, 69, 1, 35, 13, 38, 58, 9, 56, 15, 24, 100, 26, 33, 55, 10, 69, 49, 18, 65, 114, 31, 40, 55, 10, 81, 97, 31, 34, 130, 13, 36, 23, 82, 134, 11, 66, 25, 40, 15, 146, 63, 47, 107
Offset: 1

Views

Author

N. J. A. Sloane, Jan 22 2003

Keywords

Comments

It is known that a(n) >= 0.

References

  • József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 77, section III.1.1.a.
  • Wacław Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964.

Links

Crossrefs

Cf. A000203, A002808, A079528, A055682, A058208.

Programs

  • Mathematica
    s[n_] := If[CompositeQ[n], DivisorSigma[1, n] - Ceiling[n + Sqrt[n]], Nothing]; Array[s, 100] (* Amiram Eldar, Apr 25 2024 *)
  • PARI
    lista(nn) = forcomposite(n=1, nn, print1(sigma(n) - ceil(n + sqrt(n)), ", ")); \\ Michel Marcus, Dec 12 2014
  • Python
    from math import isqrt
from sympy import divisor_sigma, composite
def A079529(n): return divisor_sigma(m:=composite(n))-1-m-isqrt(m-1) # Chai Wah Wu, Jul 29 2022

A060903 a(n) = floor(6*n*sqrt(n)/Pi^2).

Original entry on oeis.org

0, 0, 1, 3, 4, 6, 8, 11, 13, 16, 19, 22, 25, 28, 31, 35, 38, 42, 46, 50, 54, 58, 62, 67, 71, 75, 80, 85, 90, 94, 99, 104, 110, 115, 120, 125, 131, 136, 142, 148, 153, 159, 165, 171, 177, 183, 189, 195, 202, 208, 214, 221, 227, 234, 241, 247, 254, 261, 268, 275, 282
Offset: 0

Views

Author

Henry Bottomley, May 05 2001

Keywords

Comments

Conjecture: the sum of the divisors of n is less than a(n) for n exceeding 12. - Robert G. Wilson v, May 14 2014

Links

Crossrefs

Cf. A000093, A000203, A000796, A058208.

Programs

  • Mathematica
    f[n_] := Floor[6 n^(3/2)/Pi^2]; Array[f, 61, 0] (* Robert G. Wilson v, May 14 2014 *)
  • PARI
    { default(realprecision, 100); t=Pi^2/6; for (n=0, 1000, write("b060903.txt", n, " ", n*sqrt(n)\t); ) } \\ Harry J. Smith, Jul 14 2009

Formula

a(n) = A000203(n) + A058208(n).
a(n) = floor(6*n^(3/2)/Pi^2).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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