%I #11 Feb 23 2023 11:36:55
%S 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,
%T 16,16,48,16,80,48,112,8,9,36,9,72,45,36,63,144,9,20,10,60,40,20,15,
%U 140,80,180,10,11,44,33,88,55,132,77,176,99,220,11
%N Triangle of numerators of harmonic mean of n and m, 1 <= m <= n.
%C 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.
%C H(n,m) = H(m,n).
%C For the rationals H(n,m)/2 see A221918(n,m)/A221919(n,m). See the comments under A221918.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean</a>.
%F a(n,m) = numerator(2*n*m/(n+m)), 1 <= m <= n.
%F a(n,m) = 2*n*m/gcd(n+m,2*n*m) = 2*n*m/gcd(n+m,2*m^2), n >= 0.
%e The triangle of numerators of H(n,m), called a(n,m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10 11 ...
%e 1: 1
%e 2: 4 2
%e 3: 3 12 3
%e 4: 8 8 24 4
%e 5: 5 20 15 40 5
%e 6: 12 3 4 24 60 6
%e 7: 7 28 21 56 35 84 7
%e 8: 16 16 48 16 80 48 112 8
%e 9: 9 36 9 72 45 36 63 144 9
%e 10: 20 10 60 40 20 15 140 80 180 10
%e 11: 11 44 33 88 55 132 77 176 99 220 11
%e ...
%e a(4,3) = numerator(24/7) = 24 = 24/gcd(7,18).
%e The triangle of the rationals H(n,m) begins:
%e n\m 1 2 3 4 5 6 7 8 9
%e 1: 1/1
%e 2: 4/3 2/1
%e 3: 3/2 12/5 3/1
%e 4: 8/5 8/3 24/7 4/1
%e 5: 5/3 20/7 15/4 40/9 5/1
%e 6: 12/7 3/1 4/1 24/5 60/11 6/1
%e 7: 7/4 28/9 21/5 56/11 35/6 84/13 7/1
%e 8: 16/9 16/5 48/11 16/3 80/13 48/7 112/15 8/1
%e 9: 9/5 36/11 9/2 72/13 45/7 36/5 63/8 144/17 9/1
%e ...
%e H(4,3) = 2*4*3/(4 + 3) = 2*4*3/7 = 24/7.
%Y Cf. A227042, A022998 (m=1), A227043 (m=2), A227106 (m=3), A227107 (m=4), A221918/A221919.
%K nonn,easy,frac,tabl
%O 1,2
%A _Wolfdieter Lang_, Jul 01 2013