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

Author

Wolfdieter Lang, Feb 21 2013

Keywords

• nonn
• easy
• tabl
• frac

Comments

The triangle of the corresponding denominators is given in A221918.

See A221918 for comments on resistance, reduced mass and radius of the twin circles in Archimedes's arbelos, as well as references.

The column sequences give A000027(n+1), A060819(n+2), A106610(n+3), A106617(n+4), A132739(n+5), A222464 for n >= m = 1,2,..., 6.

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

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.