A349084 -id:A349084 - cp's OEIS frontend

A349092 Where zeros occur in A349084. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/w + 1/x + 1/y + 1/z, 0 < w < x < y < z.

136, 252, 253, 405, 406, 465, 560, 561, 666, 703, 741, 810, 814, 818, 820, 895, 900, 901, 902, 1032, 1034, 1070, 1073, 1074, 1078, 1079, 1080, 1081, 1174, 1225, 1273, 1326, 1370, 1373, 1376, 1377, 1378, 1485, 1587, 1596, 1649, 1650, 1652, 1653, 1681, 1682, 1700, 1702, 1706, 1707, 1708, 1709, 1710, 1711, 1808

Offset: 1

Jud McCranie, Dec 12 2021

nonn

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

			136 is a term because A349084(136)=0, indicating that 16/17 = 1/w + 1/x + 1/y + 1/z has no solution.

Jud McCranie, Table of n, a(n) for n = 1..687

Cf. A349082, A002260, A003057, A349090, A349091.

A349097 Where ones occur in A349084. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/w + 1/x + 1/y + 1/z, 0 < w < x < y < z.

Original entry on oeis.org

171, 250, 325, 402, 404, 458, 463, 464, 496, 595, 660, 663, 665, 702, 817, 819, 893, 896, 899, 903, 946, 1028, 1033, 1035, 1076, 1077, 1168, 1172, 1175, 1176, 1274, 1275, 1325, 1352, 1360, 1363, 1365, 1369, 1374, 1375, 1482, 1484, 1594, 1595, 1643

Offset: 1

Jud McCranie, Dec 26 2021

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

			171 is a term because A349084(171) = 1, indicating that 18/19 = 1/w + 1/x + 1/y + 1/z has a unique solution: 1/2 + 1/3 + 1/9 + 1/342.

Cf. A349084, A002260, A003057, A349090, A349095, A349096.

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

A349092 Where zeros occur in A349084. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/w + 1/x + 1/y + 1/z, 0 < w < x < y < z.

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A349097 Where ones occur in A349084. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/w + 1/x + 1/y + 1/z, 0 < w < x < y < z.

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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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