cp's OEIS Frontend

A349084 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.

%I A349084 #31 Dec 05 2021 05:39:25
%S A349084 71,272,61,586,71,27,978,275,122,18,1591,272,71,61,17,1865,564,130,
%T A349084 145,31,18,3115,586,478,71,85,27,17,3772,1079,272,109,218,61,23,11,
%U A349084 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 A349084 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 A349084 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 A349084                     x= 1    2    3    4    5   Rationals x/y:
%C A349084 Row 1: (y=2)    71                       1/2
%C A349084 Row 2: (y=3)   272,  61                  1/3, 2/3
%C A349084 Row 3: (y=4)   586,  71,  27             1/4, 2/4, 3/4
%C A349084 Row 4: (y=5)   978, 275, 122,  18        1/5, 2/5, 3/5, 4/5
%C A349084 Row 5: (y=6)  1591, 272,  71,  61,  17   1/6, 2/6, 3/6, 4/6, 5/6
%C A349084 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 A349084 Column 1 is A241883.
%H A349084 Jud McCranie, <a href="/A349084/b349084.txt">Table of n, a(n) for n = 1..990</a>
%e A349084 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 A349084 4/5 = 1/2 + 1/4 + 1/21 + 1/420
%e A349084    = 1/2 + 1/4 + 1/22 + 1/220
%e A349084    ... 15 solutions omitted
%e A349084    = 1/3 + 1/5 + 1/6 + 1/10
%Y A349084 Cf. A002260, A003057, A349082, A349083, A241883.
%K A349084 nonn,tabl
%O A349084 1,1
%A A349084 _Jud McCranie_, Nov 11 2021