A227611 -id:A227611 - 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]

A226641 Number of ways to express 2/n as Egyptian fractions in just three terms; i.e., 2/n = 1/x + 1/y + 1/z satisfying 1<=x<=y<=z.

Original entry on oeis.org

1, 3, 8, 10, 12, 21, 17, 28, 26, 36, 25, 57, 20, 42, 81, 70, 25, 79, 32, 96, 86, 62, 42, 160, 53, 59, 89, 136, 33, 196, 37, 128, 103, 73, 185, 211, 32, 80, 160, 292, 40, 245, 40, 157, 235, 93, 60, 366, 85, 156, 147, 174, 42, 230, 223, 340, 143, 106, 76, 497, 34, 90, 331, 269, 206, 322, 50, 211, 175, 453, 72, 538, 37, 85, 332, 216, 260, 378, 69, 604, 167, 121, 79, 623, 204, 104, 203, 473, 59, 648, 253, 204, 166, 135, 318, 706, 46, 227, 427, 437

Offset: 1

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

nonn

Daniel Leary, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Egyptian fractions

Cf. A227611, A004194, A226642, A192787, A226644, A226645, A226646.

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

a[n_] := Length@ Solve[ 2/n == 1/x + 1/y + 1/z && 1 <= x <= y <= z, {x, y, z}, Integers]; Array[a, 70]

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.

Original entry on oeis.org

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

Jud McCranie, Nov 09 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)    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).

			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.

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

Cf. A002260, A003057, A349082.

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

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

A227612 Table read by antidiagonals: Number of ways m/n can be expressed as the sum of three distinct unit fractions, i.e., m/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and read by antidiagonals.

Original entry on oeis.org

1, 0, 6, 0, 1, 15, 0, 1, 5, 22, 0, 0, 1, 6, 30, 0, 0, 1, 3, 9, 45, 0, 0, 1, 1, 7, 15, 36, 0, 0, 0, 2, 2, 6, 14, 62, 0, 0, 0, 1, 1, 5, 6, 22, 69, 0, 0, 0, 1, 1, 1, 5, 16, 21, 84, 0, 0, 0, 0, 1, 1, 3, 6, 15, 30, 56, 0, 0, 0, 0, 1, 4, 1, 5, 4, 15, 22, 142, 0, 0, 0, 0, 0, 1, 1, 3, 9, 9, 13, 45, 53

Offset: 1

Robert G. Wilson v, Jul 17 2013

The main diagonal is 1, 1, 1, 1, 1, 1, 1, ..., ; i.e., 1 = 1/2 + 1/3 + 1/6.

			  m\n| 1  2   3   4   5   6   7   8   9  10  11   12  13   14   15
  ---+------------------------------------------------------------
   1 | 1  6  15  22  30  45  36  62  69  84  56  142  53  124  178  A227610
   2 | 0  1   5   6   9  15  14  22  21  30  22   45  17   36   72  A227611
   3 | 0  1   1   3   7   6   6  16  15  15  13   22   8   27   30  A075785
   4 | 0  0   1   1   2   5   5   6   4   9   7   15   4   14   33  A073101
   5 | 0  0   1   2   1   1   3   5   9   6   3   12   5   18   15  A075248
   6 | 0  0   0   1   1   1   1   3   5   7   5    6   1    6    9  n/a
   7 | 0  0   0   1   1   4   1   2   2   2   2    9   6    6    7  n/a
   8 | 0  0   0   0   1   1   1   1   1   2   0    5   3    5   15  n/a
   9 | 0  0   0   0   0   1   1   3   1   1   0    3   1    2    7  n/a
  10 | 0  0   0   0   0   1   0   2   2   1   0    1   1    3    5  n/a
.
Antidiagonals are {1}, {0, 6}, {0, 1, 15}, {0, 1, 5, 22}, {0, 0, 1, 6, 30}, {0, 0, 1, 3, 9, 45}, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
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
Oakland University, The Erdős Number Project
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A002966, A073546, A227610 (1/n), A227611 (2/n), A075785 (3/n), A073101 (4/n), A075248 (5/n).

f[m_, n_] := Length@ Solve[m/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Table[ f[n, m - n + 1], {m, 12}, {n, m, 1, -1}]

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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A226641 Number of ways to express 2/n as Egyptian fractions in just three terms; i.e., 2/n = 1/x + 1/y + 1/z satisfying 1<=x<=y<=z.

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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.

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A227612 Table read by antidiagonals: Number of ways m/n can be expressed as the sum of three distinct unit fractions, i.e., m/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and read by antidiagonals.

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