A221919 -id:A221919 - cp's OEIS frontend

A221918 Triangle of denominators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

1, 2, 1, 3, 6, 3, 4, 4, 12, 2, 5, 10, 15, 20, 5, 6, 3, 2, 12, 30, 3, 7, 14, 21, 28, 35, 42, 7, 8, 8, 24, 8, 40, 24, 56, 4, 9, 18, 9, 36, 45, 18, 63, 72, 9, 10, 5, 30, 20, 10, 15, 70, 40, 90, 5, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 11, 12, 12, 12, 3, 60, 4, 84, 24, 36, 60, 132, 6

Offset: 1

Wolfdieter Lang, Feb 21 2013

The corresponding triangle of numerators is A221919.
The law for the electrical resistance in a parallel circuit with two resistors R1 and R2 is 1/R = 1/R1 + 1/R2. Here we take 1/R(n,m) =  1/n + 1/m, with n >= m> =1, and R(n,m) = a(n,m)/A221919(n,m).
The reduced mass mu in a two body problem with masses m1 and m2 is given by 1/mu = 1/m1 + 1/m2.
The radius R of the twin circles of Archimedes' arbelos with the radii of the two small half-circles r1 and r2 is given by 1/R = 1/r1 +1/r2. The large half-circle has radius r =  r1 + r2. See, e.g., the Bankoff reference (according to which one should speak of a triple of such radius R circles). There are much more such radius R circles. See the Arbelos references given by Schoch, especially reference [3].
The columns give A000027, A145979(n-2), A221920, A221921, A222463 for m = 1, 2, ..., 5.
This and the companion entry resulted from a remark on the twin circles in Archimedes' arbelos in the Strick reference, p. 13, and the obvious question about their radii and centers. See the MathWorld link, also for more references.
The rationals R(n,m) = a(n,m)/A221919(n,m) (in lowest terms) equal H(n,m)/2, where H(n,m) = A227041(n,m)/A227042(n,m) is the harmonic mean of m and n. - Wolfdieter Lang, Jul 02 2013

			The triangle a(n,m) begins:
n\m    1    2    3    4    5    6   7   8   9   10   11  12 ...
1:     1
2:     2    1
3:     3    6    3
4:     4    4   12    2
5:     5   10   15   20    5
6:     6    3    2   12   30    3
7:     7   14   21   28   35   42   7
8:     8    8   24    8   40   24  56   4
9:     9   18    9   36   45   18  63  72   9
10:   10    5   30   20   10   15  70  40  90    5
11:   11   22   33   44   55   66  77  88  99  110   11
12:   12   12   12    3   60    4  84  24  36   60  132   6
...
a(n,1) = n because 1/R(n,1) =  1/n +1/1 = (n+1)/n, hence a(n,1) = denominator(1/n +1/1/) = n =  numerator(R(n,1)).
a(5,3) = denominator(1/5 + 1/3) = denominator(8/15 ) = 15.
a(6,3) = denominator(1/6 + 1/3) = denominator(9/18 ) = denominator(1/2) = 2.
The triangle of rationals R(n,m) = n*m/(n+m) = a(n,m)/A221919(n,m) given by 1/R(n,m) = 1/n + 1/m starts:
n\m    1    2      3     4      5     6     7     8     9   10
1:    1/2
2:    2/3    1
3:    3/4   6/5   3/2
4:    4/5   4/3  12/7    2
5:    5/6  10/7  15/8   20/9   5/2
6:    6/7   3/2    2    12/5  30/11   3
7:    7/8  14/9  21/10  28/11 35/12 42/13  7/2
8:    8/9   8/5  24/11   8/3  40/13 24/7  56/15   4
9:   9/10  18/11  9/4   36/13 45/14 18/5  63/16 72/17  9/2
10:  10/11  5/3  30/13  20/7  10/3  15/4  70/17 40/9  90/19  5
...

L. Bankoff, Are the Twin Circles of Archimedes Really Twins?, Mathematics Mag. 47,4 (1974) 214-218.
H. K. Strick, Geschichten aus der Mathematik, Spektrum der Wissenschaft - Spezial 2/2009.

T. Schoch, Arbelos References.
Eric W. Weisstein, Arbelos (MathWorld).

Cf. A221919 (companion).

a[n_, m_] := Denominator[1/n + 1/m]; Table[a[n, m], {n, 1, 12}, {m, 1, n}] // Flatten  (* Jean-François Alcover, Feb 25 2013 *)

a(n,m) = denominator(1/n +1/m) = numerator(n*m/(n+m)), n >= m >= 1 and 0 otherwise.

a(n,m)/A221919(n,m) = R(n,m) = n*m/(n+m). 1/R(n,m) = 1/n + 1/m.

A227041 Triangle of numerators of harmonic mean of n and m, 1 <= m <= n.

Original entry on oeis.org

1, 4, 2, 3, 12, 3, 8, 8, 24, 4, 5, 20, 15, 40, 5, 12, 3, 4, 24, 60, 6, 7, 28, 21, 56, 35, 84, 7, 16, 16, 48, 16, 80, 48, 112, 8, 9, 36, 9, 72, 45, 36, 63, 144, 9, 20, 10, 60, 40, 20, 15, 140, 80, 180, 10, 11, 44, 33, 88, 55, 132, 77, 176, 99, 220, 11

Offset: 1

Wolfdieter Lang, Jul 01 2013

The harmonic mean H(n,m) is the reciprocal of the arithmetic mean of the reciprocals of n and m: H(n,m) = 1/((1/2)*(1/n +1/m)) = 2*n*m/(n+m). 1/H(n,m) marks the middle of the interval [1/n, 1/m] if m < n: 1/H(n,m) = 1/n + (1/2)*(1/m - 1/n). For m < n one has m < H(n,m) < n, and H(n,n) = n.
  H(n,m) = H(m,n).
For the rationals H(n,m)/2 see A221918(n,m)/A221919(n,m). See the comments under A221918.

			The triangle of numerators 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:   4   2
3:   3  12   3
4:   8   8  24   4
5:   5  20  15  40   5
6:  12   3   4  24  60    6
7:   7  28  21  56  35   84    7
8:  16  16  48  16  80   48  112    8
9:   9  36   9  72  45   36   63  144    9
10: 20  10  60  40  20   15  140   80  180   10
11: 11  44  33  88  55  132   77  176   99  220  11
...
a(4,3) = numerator(24/7) = 24 = 24/gcd(7,18).
The triangle of the rationals H(n,m) begins:
n\m    1      2     3     4      5      6      7      8   9
1:   1/1
2:   4/3    2/1
3:   3/2   12/5   3/1
4:   8/5    8/3  24/7   4/1
5:   5/3   20/7  15/4  40/9    5/1
6:  12/7    3/1   4/1  24/5  60/11    6/1
7:   7/4   28/9  21/5 56/11   35/6  84/13    7/1
8:  16/9   16/5 48/11  16/3  80/13   48/7 112/15    8/1
9:   9/5  36/11   9/2 72/13   45/7   36/5   63/8 144/17 9/1
...
H(4,3) = 2*4*3/(4 + 3) = 2*4*3/7 = 24/7.

Eric Weisstein's World of Mathematics, Harmonic Mean.

Cf. A227042, A022998 (m=1), A227043 (m=2), A227106 (m=3), A227107 (m=4), A221918/A221919.

a(n,m) = numerator(2*n*m/(n+m)), 1 <= m <= n.

a(n,m) = 2*n*m/gcd(n+m,2*n*m) = 2*n*m/gcd(n+m,2*m^2), n >= 0.

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

Wolfdieter Lang, Jul 01 2013

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

			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.

Eric Weisstein's World of Mathematics, Harmonic Mean.

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

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.

A222464 a(n) = (n+6)/gcd(n*6,n+6), n >= 6.

Original entry on oeis.org

1, 13, 7, 5, 4, 17, 1, 19, 5, 7, 11, 23, 2, 25, 13, 3, 7, 29, 5, 31, 8, 11, 17, 35, 1, 37, 19, 13, 10, 41, 7, 43, 11, 5, 23, 47, 4, 49, 25, 17, 13, 53, 3, 55, 14, 19, 29, 59, 5, 61, 31, 7, 16, 65, 11, 67, 17, 23, 35, 71, 2, 73, 37, 25, 19, 77, 13, 79, 20, 9, 41

Offset: 6

Wolfdieter Lang, Feb 21 2013

This is the sixth column (m=6) of the triangle A221919.

			a(10) = numerator(16/60) = numerator(4/15) = 4 = 16/gcd(60,16) = 16/gcd(36,16).

Cf. A221919, A000027(n+1) (m=1), A060819(n+2) (m=2), A106610(n+3) (m=3), A106617(n+4) (m=4), A132739(n+5) (m=5).

a(n) = A221919(n,6) = denominator(n*6/(n+6)) = numerator(n+6/n*6) = (n+6)/gcd(n*6,n+6) = (n+6)/gcd(36,n+6), n >= 6.

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A221918 Triangle of denominators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.

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A227041 Triangle of numerators of harmonic mean of n and m, 1 <= m <= n.

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

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A222464 a(n) = (n+6)/gcd(n*6,n+6), n >= 6.

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