A349083 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.

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

Offset: 1

Author

Jud McCranie, Nov 09 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) 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).

Examples

			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.
		

Links

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

Crossrefs

Cf. A002260, A003057, A349082.

Columns x=1..5: A227610, A227611, A075785, A073101, A075248.

Programs

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" && qCharles R Greathouse IV, Nov 09 2021