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%I #42 Nov 24 2021 20:10:03
%S 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,
%T 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,
%U 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
%N 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.
%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): 1 1/2
%C Row 2 (y=3): 1, 1 1/3, 2/3
%C Row 3 (y=4): 2, 1, 1 1/4, 2/4, 3/4
%C Row 4 (y=5): 1, 1, 1, 0 1/5, 2/5, 3/5, 4/5
%C Row 5 (y=6): 4, 1, 1, 1, 1 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, ... 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).
%C A018892 is a subsequence (for x/y = 1/n).
%H Jud McCranie, <a href="/A349082/b349082.txt">Table of n, a(n) for n = 1..990</a>
%e The fourth rational number is 1/4, 1/4 = 1/5 + 1/20 = 1/6 + 1/12, so a(4)=2.
%Y Cf. A002260, A003057.
%Y Columns: A018892 (x=1), A046079 (x=2).
%K nonn,tabl
%O 1,4
%A _Jud McCranie_, Nov 07 2021
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