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A349083
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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.
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6
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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, 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, 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
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OFFSET
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1,1
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COMMENTS
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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:
x = 1 2 3 4 5 Rationals x/y:
Row 1: (y=2) 6 1/2
Row 2: (y=3) 15, 5 1/3, 2/3
Row 3: (y=4) 22, 6, 3 1/4, 2/4, 3/4
Row 4: (y=5) 30, 9, 7, 2 1/5, 2/5, 3/5, 4/5
Row 5: (y=6) 45, 15, 6, 5, 1 1/6, 2/6, 3/6, 4/6, 5/6
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).
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LINKS
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EXAMPLE
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The sixth rational number is 3/4;
3/4 = 1/2 + 1/5 + 1/20
= 1/2 + 1/6 + 1/12
= 1/3 + 1/4 + 1/5,
so a(6)=3.
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PROG
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(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))
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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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