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

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

Author

Wolfdieter Lang, Jul 01 2013

Keywords

• nonn
• easy
• frac
• tabl

Comments

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Harmonic Mean.

Crossrefs

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

Formula

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.