A347566 -id:A347566 - cp's OEIS frontend

A227610 Number of ways 1/n can be expressed as the sum of three distinct unit fractions: 1/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.

1, 6, 15, 22, 30, 45, 36, 62, 69, 84, 56, 142, 53, 124, 178, 118, 67, 191, 74, 274, 227, 145, 87, 342, 146, 162, 216, 322, 100, 461, 84, 257, 304, 199, 435, 508, 79, 204, 360, 580, 115, 587, 98, 455, 618, 192, 129, 676, 217, 417, 369, 449, 119, 573, 543, 759, 367, 240, 166, 1236, 102, 261, 857, 428, 568, 717, 115, 537, 460, 1018, 155, 1126, 112, 276, 839

Offset: 1

Robert G. Wilson v, Jul 17 2013

nonn

See A073101 for the 4/n conjecture due to Erdős and Straus.

			a(1)=1 because 1 = 1/2 + 1/3 + 1/6;
a(2)=6 because 1/2 = 1/3 + 1/7 + 1/42 = 1/3 + 1/8 + 1/24 = 1/3 + 1/9 + 1/18 = 1/3 + 1/10 + 1/15 = 1/4 + 1/5 + 1/20 = 1/4 + 1/6 + 1/12;
a(3)=15 because 1/3 = 1/x + 1/y + 1/z presented as {x,y,z}: {4,13,156}, {4,14,84}, {4,15,60}, {4,16,48}, {4,18,36}, {4,20,30}, {4,21,28}, {5,8,120}, {5,9,45}, {5,10,30}, {5,12,20}, {6,7,42}, {6,8,24}, {6,9,18}, {6,10,15}; etc.

Jud McCranie, Table of n, a(n) for n = 1..500
Christian Elsholtz, Sums Of k Unit Fractions, Trans. Amer. Math. Soc. 353 (2001), 3209-3227.
David Eppstein, Algorithms for Egyptian Fractions
David Eppstein, Ten Algorithms for Egyptian Fractions, Wolfram Library Archive.
Ron Knott, Egyptian Fractions
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A002966, A073546.
Cf. A227611 (2/n), A075785 (3/n), A073101 (4/n), A075248 (5/n), A227612.
Cf. A347566, A347569.

f[n_] := Length@ Solve[1/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Array[f, 70]

A075785 Number of ways to express 3/n as Egyptian fractions in just three terms.

Original entry on oeis.org

0, 1, 1, 3, 7, 6, 6, 16, 15, 15, 13, 22, 8, 27, 30, 26, 21, 45, 8, 59, 36, 29, 20, 62, 32, 45, 69, 67, 27, 84, 14, 59, 56, 44, 129, 142, 15, 45, 53, 130, 31, 124, 21, 131, 178, 40, 29, 118, 38, 88, 67, 102, 43, 191, 102, 180, 74, 57, 43, 274, 21, 75, 227, 86, 144, 145, 23, 121, 87

Offset: 1

Robert G. Wilson v, Aug 18 2002

nonn

Christian Elsholtz, Sums Of k Unit Fractions
David Eppstein, Algorithms for Egyptian Fractions

Cf. A073101, A347566, A347569.

Needs["MyOwn`Egypt`"]; Table[ Length[ EgyptianFraction[3/n, Method -> Lexicographic, MaxTerms -> 3, MinTerms -> 3, Duplicates -> Disallow, OutputFormat -> Plain]], {n, 5, 70}]
f[n_] := Length@ Solve[3/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Array[f, 69] (* Robert G. Wilson v, Jul 17 2013 *)

A347569 Number of ways 1/n can be expressed as the sum of six distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t + 1/u, with 0 < p < q < r < s < t < u.

Original entry on oeis.org

2320, 244817, 3421052, 18420699, 64025680, 131223239, 431008820, 681922142

Offset: 1

Jud McCranie, Sep 06 2021

Cf. A002966, A073546, A227610, A241883, A347566.

a(7) and a(8) from Jud McCranie, Oct 15 2021

A349085 The number of five-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,t) such that x/y = 1/p + 1/q + 1/r + 1/s + 1/t where p, q, r, s, and t are integers with p < q < r < s < t.

Original entry on oeis.org

2293, 15304, 1890, 47314, 2293, 662, 112535, 19311, 6650, 510, 190665, 15304, 2293, 1890, 298, 368474, 64992, 10447, 11362, 1666, 708, 577623, 47314, 44843, 2293, 3820, 662, 489, 925336, 147545, 15304, 5302, 18606, 1890, 850, 277, 1164976, 112535, 39798, 19311, 2293, 6650, 1152

Offset: 1

Jud McCranie, Nov 13 2021

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)    2293                            1/2
Row 2: (y=3)   15304,  1890                     1/3, 2/3
Row 3: (y=4)   47314,  2293,  662               1/4, 2/4, 3/4
Row 4: (y=5)  112535, 19311, 6650,  510         1/5, 2/5, 3/5, 4/5
Row 5: (y=6)  190665, 15304, 2293, 1890, 298    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 A347566, skipping the first term.

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

Cf. A241883, A002260, A003057, A349082, A349083, A349084.

cp's OEIS Frontend

A227610 Number of ways 1/n can be expressed as the sum of three distinct unit fractions: 1/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.

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A075785 Number of ways to express 3/n as Egyptian fractions in just three terms.

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A347569 Number of ways 1/n can be expressed as the sum of six distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t + 1/u, with 0 < p < q < r < s < t < u.

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A349085 The number of five-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,t) such that x/y = 1/p + 1/q + 1/r + 1/s + 1/t where p, q, r, s, and t are integers with p < q < r < s < t.

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