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.

71, 272, 61, 586, 71, 27, 978, 275, 122, 18, 1591, 272, 71, 61, 17, 1865, 564, 130, 145, 31, 18, 3115, 586, 478, 71, 85, 27, 17, 3772, 1079, 272, 109, 218, 61, 23, 11, 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

Offset: 1

Author

Jud McCranie, Nov 11 2021

Keywords

• nonn
• tabl

Comments

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) 71 1/2

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

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

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

Row 5: (y=6) 1591, 272, 71, 61, 17 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).

Column 1 is A241883.

Examples

			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:
4/5 = 1/2 + 1/4 + 1/21 + 1/420
   = 1/2 + 1/4 + 1/22 + 1/220
   ... 15 solutions omitted
   = 1/3 + 1/5 + 1/6 + 1/10
		

Links

• Jud McCranie, Table of n, a(n) for n = 1..990

Crossrefs

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