A349083 -id:A349083 - cp's OEIS frontend

A349091 Where zeros occur in A349083. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/x + 1/y + 1/z, 0 < x < y < z.

53, 54, 55, 78, 91, 120, 128, 134, 135, 136, 162, 167, 168, 170, 171, 210, 226, 228, 230, 231, 246, 247, 248, 249, 250, 251, 252, 253, 288, 298, 299, 300, 319, 321, 323, 324, 325, 345, 347, 350, 351, 377, 378, 390, 392, 396, 397, 398, 399, 401, 402, 403, 404, 405, 406, 435, 447, 450, 453, 457, 458, 459, 460, 462, 463, 464, 465

Offset: 1

Jud McCranie, Dec 12 2021

nonn

For index k, p/q = A002260(k)/A003057(k).

			53 is a term because A349083(53)=0, indicating that 8/11 = 1/x + 1/y + 1/z has no solution.

Jud McCranie, Table of n, a(n) for n = 1..732

Cf. A349082, A002260, A003057, A349090, A349092.

A349096 Where ones occur in A349083. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/x + 1/y + 1/z, 0 < x < y < z.

Original entry on oeis.org

15, 21, 36, 45, 65, 72, 75, 76, 77, 89, 90, 105, 118, 131, 132, 133, 151, 152, 153, 165, 166, 169, 189, 190, 206, 207, 208, 209, 225, 227, 229, 241, 242, 245, 273, 276, 292, 293, 294, 295, 297, 312, 317, 318, 320, 322, 348, 349, 373, 374, 375, 376, 387, 391, 400, 431

Offset: 1

Jud McCranie, Dec 26 2021

nonn

For index k, p/q = A002260(k)/A003057(k).

			15 is a term because A349083(15)=1, indicating that 5/6 = 1/x + 1/y + 1/z has a unique solution: 1/2 + 1/4 + 1/12.

Cf. A349083, A002260, A003057, A349090, A349095.

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.

Original entry on oeis.org

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.

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.

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

A349091 Where zeros occur in A349083. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/x + 1/y + 1/z, 0 < x < y < z.

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A349096 Where ones occur in A349083. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/x + 1/y + 1/z, 0 < x < y < z.

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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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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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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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