OFFSET
1,2
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).
LINKS
Eric Weisstein's World of Mathematics, Harmonic Mean.
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.
EXAMPLE
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.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Jul 01 2013
STATUS
approved