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A227042
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Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.
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7
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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
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OFFSET
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1,2
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COMMENTS
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See the comments under A227041. a(n,m) gives the denominator of H(n,m) = 2*n*m/(n+m) in lowest terms.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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