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A349083 The number of three-term Egyptian fractions of rational numbers x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q,r) such that x/y = 1/p + 1/q + 1/r where p, q, and r are integers with p < q < r.

%I A349083 #37 Dec 05 2021 05:39:17
%S A349083 6,15,5,22,6,3,30,9,7,2,45,15,6,5,1,36,14,6,5,3,1,62,22,16,6,5,3,2,69,
%T A349083 21,15,4,9,5,2,1,84,30,15,9,6,7,2,2,1,56,22,13,7,3,5,2,0,0,0,142,45,
%U A349083 22,15,12,6,9,5,3,1,2,53,17,8,4,5,1,6,3,1,1,1,0,124,36,27,14,18,6,6,5,2,3,1,1,0
%N A349083 The number of three-term Egyptian fractions of rational numbers x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q,r) such that x/y = 1/p + 1/q + 1/r where p, q, and r are integers with p < q < r.
%C A349083 The sequence are the terms in a triangle, where the rows correspond to the denominator of the rational number (starting with row 2, column 1) and the columns correspond to the numerators:
%C A349083             x = 1   2   3   4   5   Rationals x/y:
%C A349083 Row 1: (y=2)    6                   1/2
%C A349083 Row 2: (y=3)   15,  5               1/3, 2/3
%C A349083 Row 3: (y=4)   22,  6,  3           1/4, 2/4, 3/4
%C A349083 Row 4: (y=5)   30,  9,  7,  2       1/5, 2/5, 3/5, 4/5
%C A349083 Row 5: (y=6)   45, 15,  6,  5,  1   1/6, 2/6, 3/6, 4/6, 5/6
%C A349083 Alternatively, order the rational numbers, x/y, 0 < x/y < 1, in this order: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, ... The numerators of the n-th rational number are A002260(n) and the denominators are A003057(n).
%H A349083 Jud McCranie, <a href="/A349083/b349083.txt">Table of n, a(n) for n = 1..990</a>
%e A349083 The sixth rational number is 3/4;
%e A349083   3/4 = 1/2 + 1/5 + 1/20
%e A349083       = 1/2 + 1/6 + 1/12
%e A349083       = 1/3 + 1/4 + 1/5,
%e A349083 so a(6)=3.
%o A349083 (PARI) Efrac3(x,y)=sum(p=if(y%x,y\x,y\x+1),3*y\x, my(N=x/y-1/p); sum(q=max(if(numerator(N)==1,1\N+1,1\N),p+1),2\N, my(M=N-1/q,r=1/M); type(r)=="t_INT" && q<r))
%o A349083 row(n)=vector(n-1,x,Efrac3(x,n)) \\ _Charles R Greathouse IV_, Nov 09 2021
%Y A349083 Cf. A002260, A003057, A349082.
%Y A349083 Columns x=1..5: A227610, A227611, A075785, A073101, A075248.
%K A349083 nonn,tabl
%O A349083 1,1
%A A349083 _Jud McCranie_, Nov 09 2021