A227612 -id:A227612 - 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]

A226646 Number of ways to express m/n as Egyptian fractions in just three terms, that is, m/n = 1/x + 1/y + 1/z satisfying 1 <= x <= y <= z and read by antidiagonals.

Original entry on oeis.org

3, 1, 10, 1, 3, 21, 0, 3, 8, 28, 0, 1, 3, 10, 36, 0, 1, 3, 6, 12, 57, 0, 1, 2, 3, 10, 21, 42, 0, 0, 1, 4, 2, 10, 17, 70, 0, 0, 1, 3, 3, 8, 9, 28, 79, 0, 0, 0, 1, 3, 4, 7, 20, 26, 96, 0, 0, 1, 1, 2, 3, 4, 10, 21, 36, 62, 0, 0, 0, 1, 1, 7, 1, 7, 6, 21, 25, 160, 0, 0, 0, 1, 0, 3, 3, 6, 12, 12, 16, 57, 59

Offset: 1

Allan C. Wechsler and Robert G. Wilson v, Aug 17 2013

See A073101 for the 4/n conjecture due to Erdös and Straus.
The first upper diagonal is 10, 8, 6, 2, 4, 1, 2, 1, 2, 0, 3, 0, 0, 1, 0, 0, 1, 0, 1, 0,... .
The main diagonal is: 3, 3, 3, 3, 3, 3, ... since 1 = 1/2 + 1/3 + 1/6 = 1/2 + 1/4 + 1/4 = 1/3 + 1/3 + 1/3. See A002966(3).
The first lower diagonal is 1, 3, 3, 4, 3, 7, 3, 5, 4, 6, 3, 10, 3, 6, 6, 6, 3, 9, 3, 9, ... .
The antidiagonal sum is 3, 11, 25, 39, 50, 79, 79, 104, 131, 157, 140, 229, 169, 220, 295, 282, ... .

			../n
m/ 1...2...3...4...5...6...7...8...9..10..11...12..13...14...15 =Allocation nbr.
.1 3..10..21..28..36..57..42..70..79..96..62..160..59..136..196 A004194
.2 1...3...8..10..12..21..17..28..26..36..25...57..20...42...81 A226641
.3 1...3...3...6..10..10...9..20..21..21..16...28..11...33...36 A226642
.4 0...1...3...3...2...8...7..10...6..12...9...21...4...17...39 A192787
.5 0...1...2...4...3...4...4...7..12..10...3...17...6...21...21 A226644
.6 0...1...1...3...3...3...1...6...8..10...7...10...1....9...12 A226645
.7 0...0...1...1...2...7...3...2...3...5...2...13...8...10....9 n/a
.8 0...0...0...1...1...3...3...3...1...2...0....8...3....7...19 n/a
.9 0...0...1...1...0...3...2...5...3...2...0....6...2....4...10 n/a
10 0...0...0...1...1...2...0...4...4...3...0....4...1....4....8 n/a
Triangle (by antidiagonals):
  {3},
  {1, 10},
  {1, 3, 21},
  {0, 3, 8, 28},
  {0, 1, 3, 10, 36},
  {0, 1, 3, 6, 12, 57},
  ...

Christian Elsholtz, Sums Of k Unit Fractions
David Eppstein, Algorithms for Egyptian Fractions
David Eppstein, Ten Algorithms for Egyptian Fractions
Ron Knott, Egyptian Fractions
Oakland University, The Erdős Number Project
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A227612, A226640, A226641, A226642, A192787, A226644, A226645.

f[m_, n_] := Length@ Solve[m/n == 1/x + 1/y + 1/z && 1 <= x <= y <= z, {x, y, z}, Integers]; Table[f[n, m - n + 1], {m, 12}, {n, m, 1, -1}] // Flatten

A227611 Number of ways 2/n can be expressed as the sum of three distinct unit fractions: 2/n = 1/x + 1/y + 1/z with 0 < x < y < z.

Original entry on oeis.org

0, 1, 5, 6, 9, 15, 14, 22, 21, 30, 22, 45, 17, 36, 72, 62, 22, 69, 29, 84, 77, 56, 39, 142, 48, 53, 82, 124, 30, 178, 34, 118, 94, 67, 176, 191, 29, 74, 151, 274, 37, 227, 37, 145, 220, 87, 57, 342, 80, 146, 138, 162, 39, 216, 214, 322, 134, 100, 73, 461, 31, 84, 316, 257, 197, 304, 47, 199, 166, 435, 69, 508, 34, 79, 317

Offset: 1

Robert G. Wilson v, Jul 17 2013

nonn

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

Cf. A002966, A073546.

Cf. A227610 (1/n), A075785 (3/n), A073101 (4/n), A075248 (5/n), A227612.

f[n_] := Length@ Solve[2/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Array[f, 75]

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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A226646 Number of ways to express m/n as Egyptian fractions in just three terms, that is, m/n = 1/x + 1/y + 1/z satisfying 1 <= x <= y <= z and read by antidiagonals.

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A227611 Number of ways 2/n can be expressed as the sum of three distinct unit fractions: 2/n = 1/x + 1/y + 1/z with 0 < x < y < z.

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Mathematica