A241883 -id:A241883 - cp's OEIS frontend

A349084 The number of four-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) such that x/y = 1/p + 1/q + 1/r + 1/s where p, q, r, and s are integers with p < q < r < s.

71, 272, 61, 586, 71, 27, 978, 275, 122, 18, 1591, 272, 71, 61, 17, 1865, 564, 130, 145, 31, 18, 3115, 586, 478, 71, 85, 27, 17, 3772, 1079, 272, 109, 218, 61, 23, 11, 4964, 978, 461, 275, 71, 122, 39, 18, 9, 4225, 1208, 641, 400, 59, 174, 37, 16, 5, 3, 8433, 1591, 586, 272, 214, 71, 172, 61, 27, 17, 12

Offset: 1

Jud McCranie, Nov 11 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)    71                       1/2
Row 2: (y=3)   272,  61                  1/3, 2/3
Row 3: (y=4)   586,  71,  27             1/4, 2/4, 3/4
Row 4: (y=5)   978, 275, 122,  18        1/5, 2/5, 3/5, 4/5
Row 5: (y=6)  1591, 272,  71,  61,  17   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 A241883.

			The 10th rational number under this ordering is 4/5; 4/5 has 18 representations as the sum of four distinct unit fractions, so a(10) = 18:
4/5 = 1/2 + 1/4 + 1/21 + 1/420
   = 1/2 + 1/4 + 1/22 + 1/220
   ... 15 solutions omitted
   = 1/3 + 1/5 + 1/6 + 1/10

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

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

A343074 Number of partitions of 1/n into n distinct reciprocals of positive integers.

Original entry on oeis.org

1, 1, 15, 586, 112535, 131223239

Offset: 1

Ilya Gutkovskiy, Apr 04 2021

			a(2) = 1 because we have 1/2 = 1/3 + 1/6.

Index entries for sequences related to Egyptian fractions

Cf. A006585, A063647, A227610, A241883, A342267.

a(5) from Bert Dobbelaere, Apr 05 2021

a(6) from Jud McCranie, Sep 03 2021

A347566 Number of ways 1/n can be expressed as the sum of five distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t, with 0 < p < q < r < s < t.

Original entry on oeis.org

72, 2293, 15304, 47314, 112535, 190665, 368474, 577623, 925336, 1164976, 1478492, 2051830, 2240745, 3789424, 4989958, 4672559, 4467275, 7104589, 6548335, 13844524, 13580094, 10633142, 10451326, 20262957, 16621976, 18697914, 25613090, 27523671

Offset: 1

Jud McCranie, Sep 06 2021

nonn

Cf. A002966, A073546, A227610, A241883, A347569.

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.

A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 5, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 5, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 7, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 2, 3

Offset: 1

Lars Blomberg, Feb 14 2022

nonn

By symmetry, if (i, j) is a solution then so is (j, i). When j=i we get n = 3i, corresponding to the solution 1/2 + 1/2 = 1. Therefore, when 3|n, a(n) > 0 and odd, otherwise a(n) >= 0 and even.
For n < 10^6, the largest term is a(720720) = 285, and 483188 terms are 0.
Other record terms: a(1081080) = 369, a(2162160) = 457, a(3243240) = 481, a(4324320) = 533, a(5405400) = 559, a(6126120) = 597. Record terms with index >= 360360 appear to occur at indices that are multiples of 180180. - Chai Wah Wu, Feb 15 2022

			For n = 3: (i, j) = (1, 1), so a(3) = 1. (1/2 + 1/2 = 1)
For n = 18: (i, j) = (3, 8), (6, 6), (8, 3), so a(18) = 3. (3/15 + 8/10 = 1/5 + 4/5 = 1)
For n = 20: (i, j) = (5, 8), (8, 5), so a(20) = 2.
For n = 36: (i, j) = (6, 16), (8, 15), (12, 12), (15, 8), (16, 6), so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A018892, A016017, A051908, A073546, A227610, A238795, A241883, A297895, A303400.

Cf. A351694, A351695 for records.

a(n)={my(x=n^2, y=2*n); sum(i=1,(n-1)/2, x-=2*n; y-=3; if(x%y==0,1,0))}

from sympy.abc import i, j
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A351532(n):
    return sum(1 for d in diop_quadratic(n**2+3*i*j-2*n*(i+j)) if 0 < d[0] < n and 0 < d[1] < n) # Chai Wah Wu, Feb 15 2022

The original equation can be solved for j giving j = (n(n - 2i)) / (2n - 3i). Varying i from 1 to n-1, a(n) is given by the number of integer values of j, 0 < j < n.

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

A349086 The number of six-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,u) such that x/y = 1/p + 1/q + 1/r + 1/s + 1/t + 1/u where p, q, r, s, t, and u are integers with p < q < r < s < t < u.

Original entry on oeis.org

244817, 3421052, 206917, 18420699, 244817, 49938, 64025680, 6462507, 1434759, 41993, 131223239, 3421052, 244817, 206917, 16018, 431008820, 38282319, 3506679, 3879468, 323772, 108276, 681922142, 18420699, 21874941, 244817, 659687, 49938, 45169

Offset: 1

Jud McCranie, Nov 19 2021

The sequence are the terms in a triangle, where the rows correspond to the denominator of the rational number (starting with 2) and the columns correspond to the numerators:
                  x = 1       2       3      4     5   Rationals x/y:
Row 1: (y=2)     244817                                1/2
Row 2: (y=3)    3421052  206917                        1/3, 2/3
Row 3: (y=4)   18420699  244817   49938                1/4, 2/4, 3/4
Row 4: (y=5)   64025680 6462507 1434759  41993         1/5, 2/5, 3/5, 4/5
Row 5: (y=6)  131223239 3421052  244817 206917 16018   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).

Jud McCranie, Table of n, a(n) for n = 1..36

Cf. A241883, A002260, A003057, A349082, A349083, A349084, A349085.

cp's OEIS Frontend

A349084 The number of four-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) such that x/y = 1/p + 1/q + 1/r + 1/s where p, q, r, and s are integers with p < q < r < s.

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A343074 Number of partitions of 1/n into n distinct reciprocals of positive integers.

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A347566 Number of ways 1/n can be expressed as the sum of five distinct unit fractions: 1/n = 1/p + 1/q + 1/r + 1/s + 1/t, with 0 < p < q < r < s < t.

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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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A351532 Number of integer pairs (i, j), 0 < i, j < n, such that i/(n - i) + j/(n - j) = 1.

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A349086 The number of six-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,u) such that x/y = 1/p + 1/q + 1/r + 1/s + 1/t + 1/u where p, q, r, s, t, and u are integers with p < q < r < s < t < u.

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