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%I A221919 #18 May 25 2019 22:09:49
%S A221919 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,
%T A221919 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,
%U A221919 15,16,17,18,19,20,21,2,13,7,5,1,17,1,19,5,7,11,23,1
%N A221919 Triangle of numerators of sum of two unit fractions: 1/n + 1/m, n >= m >= 1.
%C A221919 The triangle of the corresponding denominators is given in A221918.
%C A221919 See A221918 for comments on resistance, reduced mass and radius of the twin circles in Archimedes's arbelos, as well as references.
%C A221919 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.
%F A221919 a(n,m) = numerator(2/n + 1/m), n >= m >= 1, and 0 otherwise.
%F A221919 A221918(n,m)/a(n,m) = R(n,m) = n*m/(n+m). 1/R(n,m) = 1/n + 1/m.
%e A221919 The triangle a(n,m) begins:
%e A221919 n\m   1   2    3   4    5   6    7    8   9   10  11  12 ...
%e A221919 1:    2
%e A221919 2:    3   1
%e A221919 3:    4   5    2
%e A221919 4:    5   3    7   1
%e A221919 5:    6   7    8   9    2
%e A221919 6:    7   2    1   5   11   1
%e A221919 7:    8   9   10  11   12  13    2
%e A221919 8:    9   5   11   3   13   7   15    1
%e A221919 9:   10  11    4  13   14   5   16   17
%e A221919 10:  11   3   13   7    3   4   17    9  19    1
%e A221919 11:  12  13   14  15   16  17   18   15  20   21   2
%e A221919 12:  13   7    5   1   17   1   19    5   7   11  23   1
%e A221919 ...
%e A221919 a(n,1) = n + 1 because R(n,1) = n/(n+1), gcd(n,n+1) = 1, hence denominator(R(n,m)) = n + 1.
%e A221919 a(5,4) = 9 because R(5,4) = 20/9, gcd(20,9) = 1, hence denominator( R(5,4)) = 9.
%e A221919 a(6,3) = 1 because R(6,3) = 18/9 = 2/1.
%e A221919 For the rationals R(n,m) see A221918.
%t A221919 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 *)
%Y A221919 Cf. A221918 (companion triangle).
%K A221919 nonn,easy,tabl,frac
%O A221919 1,1
%A A221919 _Wolfdieter Lang_, Feb 21 2013

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