A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.
2, 3, 1, 4, 5, 2, 5, 3, 7, 1, 6, 7, 8, 9, 2, 7, 2, 1, 5, 11, 1, 8, 9, 10, 11, 12, 13, 2, 9, 5, 11, 3, 13, 7, 15, 1, 10, 11, 4, 13, 14, 5, 16, 17, 2, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1
Offset: 1
Examples
The triangle a(n,m) begins: n\m 1 2 3 4 5 6 7 8 9 10 11 12 ... 1: 2 2: 3 1 3: 4 5 2 4: 5 3 7 1 5: 6 7 8 9 2 6: 7 2 1 5 11 1 7: 8 9 10 11 12 13 2 8: 9 5 11 3 13 7 15 1 9: 10 11 4 13 14 5 16 17 10: 11 3 13 7 3 4 17 9 19 1 11: 12 13 14 15 16 17 18 15 20 21 2 12: 13 7 5 1 17 1 19 5 7 11 23 1 ... a(n,1) = n + 1 because R(n,1) = n/(n+1), gcd(n,n+1) = 1, hence denominator(R(n,m)) = n + 1. a(5,4) = 9 because R(5,4) = 20/9, gcd(20,9) = 1, hence denominator( R(5,4)) = 9. a(6,3) = 1 because R(6,3) = 18/9 = 2/1. For the rationals R(n,m) see A221918.
Crossrefs
Cf. A221918 (companion triangle).
Programs
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Mathematica
a[n_, m_] := Numerator[1/n + 1/m]; Table[a[n, m], {n, 1, 12}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 25 2013 *)
Formula
a(n,m) = numerator(2/n + 1/m), n >= m >= 1, and 0 otherwise.
A221918(n,m)/a(n,m) = R(n,m) = n*m/(n+m). 1/R(n,m) = 1/n + 1/m.
Comments