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The number of four-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q,r,s) such that x/y = 1/p + 1/q + 1/r + 1/s where p, q, r, and s are integers with p < q < r < s.
5

%I #31 Dec 05 2021 05:39:25

%S 71,272,61,586,71,27,978,275,122,18,1591,272,71,61,17,1865,564,130,

%T 145,31,18,3115,586,478,71,85,27,17,3772,1079,272,109,218,61,23,11,

%U 4964,978,461,275,71,122,39,18,9,4225,1208,641,400,59,174,37,16,5,3,8433,1591,586,272,214,71,172,61,27,17,12

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

%C 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 x= 1 2 3 4 5 Rationals x/y:

%C Row 1: (y=2) 71 1/2

%C Row 2: (y=3) 272, 61 1/3, 2/3

%C Row 3: (y=4) 586, 71, 27 1/4, 2/4, 3/4

%C Row 4: (y=5) 978, 275, 122, 18 1/5, 2/5, 3/5, 4/5

%C Row 5: (y=6) 1591, 272, 71, 61, 17 1/6, 2/6, 3/6, 4/6, 5/6

%C 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).

%C Column 1 is A241883.

%H Jud McCranie, <a href="/A349084/b349084.txt">Table of n, a(n) for n = 1..990</a>

%e The 10th rational number under this ordering is 4/5; 4/5 has 18 representations as the sum of four distinct unit fractions, so a(10) = 18:

%e 4/5 = 1/2 + 1/4 + 1/21 + 1/420

%e = 1/2 + 1/4 + 1/22 + 1/220

%e ... 15 solutions omitted

%e = 1/3 + 1/5 + 1/6 + 1/10

%Y Cf. A002260, A003057, A349082, A349083, A241883.

%K nonn,tabl

%O 1,1

%A _Jud McCranie_, Nov 11 2021