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.

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A227042 Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.

Original entry on oeis.org

1, 3, 1, 2, 5, 1, 5, 3, 7, 1, 3, 7, 4, 9, 1, 7, 1, 1, 5, 11, 1, 4, 9, 5, 11, 6, 13, 1, 9, 5, 11, 3, 13, 7, 15, 1, 5, 11, 2, 13, 7, 5, 8, 17, 1, 11, 3, 13, 7, 3, 2, 17, 9, 19, 1, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 1
Offset: 1

Views

Author

Wolfdieter Lang, Jul 01 2013

Keywords

Comments

See the comments under A227041. a(n,m) gives the denominator of H(n,m) = 2*n*m/(n+m) in lowest terms.

Examples

			The triangle of denominators of H(n,m), called a(n,m) begins:
n\m  1   2   3   4   5    6    7    8    9   10  11 ...
1:   1
2:   3   1
3:   2   5   1
4:   5   3   7   1
5:   3   7   4   9   1
6:   7   1   1   5  11    1
7:   4   9   5  11   6   13    1
8;   9   5  11   3  13    7   15    1
9:   5  11   2  13   7    5    8   17    1
10: 11   3  13   7   3    2   17    9   19    1
11:  6  13   7  15   8   17    9   19   10   21   1
...
For the triangle of the rationals H(n,m) see the example section of A227041.
H(4,2) = denominator(16/6) = denominator(8/3) = 3 = 6/gcd(6,8) = 6/2.

Links

Crossrefs

Cf. A227041, A026741 (column m=1), A000265 (m=2), A106619 (m=3), A227140(n+8) (m=4), A227108 (m=5), A221918/A221919.

Formula

a(n,m) = denominator(2*n*m/(n+m)), 1 <= m <= n.
a(n,m) = (n+m)/gcd(2*n*m, n+m) = (n+m)/gcd(n+m, 2*m^2), 1 <= m <= n.

A227109 Numerators of harmonic mean H(n, 5), n >= 0.

Original entry on oeis.org

0, 5, 20, 15, 40, 5, 60, 35, 80, 45, 20, 55, 120, 65, 140, 15, 160, 85, 180, 95, 8, 105, 220, 115, 240, 25, 260, 135, 280, 145, 60, 155, 320, 165, 340, 35, 360, 185, 380, 195, 80, 205, 420, 215, 440, 9, 460, 235, 480, 245, 100, 255, 520, 265, 540, 55, 560
Offset: 0

Views

Author

Wolfdieter Lang, Jul 01 2013

Keywords

Comments

a(n) = numerator(H(n, 5)) = numerator(10*n/(n+5)), n>=0, with H(n, 5) the harmonic mean of n and 5.
The corresponding denominator is given in A227108(n), n>= 0.
a(n+5), n>=0, is the fifth column (m=5) of the triangle A227041.

Examples

			The rationals H(n,5) begin: 0, 5/3, 20/7, 15/4, 40/9, 5, 60/11, 35/6, 80/13, 45/7, 20/3, 55/8, 120/17, 65/9, ...

Crossrefs

Cf. A227041(n+5,5), A227108 (denominator).

Programs

  • Magma
    [Numerator(10*n/(n+5)): n in [0..60]]; // Vincenzo Librandi, Nov 06 2016
  • Mathematica
    Table[Numerator[(10 n / (n + 5))], {n, 0, 60}] (* Vincenzo Librandi, Nov 06 2016 *)
  • PARI
    a(n) = numerator(10*n/(n+5)); \\ Michel Marcus, Nov 06 2016

Formula

a(n) = numerator(10*n/(n+5)), n >= 0.
a(n) = 10*n/gcd(n+5,10*n) = 10*n/gcd(n+5,50), n >= 0.
Showing 1-2 of 2 results.

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

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