A349082 The number of two-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q) pairs such that x/y = 1/p + 1/q where p and q are integers with p < q.

1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 4, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 7, 4, 2, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 3, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 4, 4, 1, 3, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 4, 3, 2, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 0, 0

Offset: 1

Author

Jud McCranie, Nov 07 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): 1 1/2

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

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

Row 4 (y=5): 1, 1, 1, 0 1/5, 2/5, 3/5, 4/5

Row 5 (y=6): 4, 1, 1, 1, 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, ... For example, in this ordering, the sixth rational number is 3/4. The numerators of the n-th rational number are A002260(n) and the denominators are A003057(n).

A018892 is a subsequence (for x/y = 1/n).

Examples

			The fourth rational number is 1/4, 1/4 = 1/5 + 1/20 = 1/6 + 1/12, so a(4)=2.
		

Links

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

Crossrefs

Cf. A002260, A003057.

Columns: A018892 (x=1), A046079 (x=2).