A349085 -id:A349085 - cp's OEIS frontend

A349098 Where ones occur in A349085. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/v + 1/w + 1/x + 1/y + 1/z, 0 < v < w < x < y < z.

2211, 3402, 3403, 3912, 3914, 3916, 4656, 5048, 5565, 5867, 6326, 8505, 8507, 8910, 9293, 9294, 9313, 9578, 9586, 9587, 9588, 10585, 11002, 11017, 11308, 11317, 11322, 11324, 11935, 12242, 12244, 12245, 12558, 12560, 13193, 13194, 13692, 13694, 13834

Offset: 1

Jud McCranie, Dec 31 2021

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

			11002 is a term because A349085(11002) = 1, indicating that 124/149 = 1/v + 1/w + 1/x + 1/y + 1/z has a unique solution, namely 1/2 + 1/4 + 1/13 + 1/189 + 1/1464372.

Cf. A349085, A002260, A003057, A349090, A349095, A349096, A349097.

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

A349098 Where ones occur in A349085. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/v + 1/w + 1/x + 1/y + 1/z, 0 < v < w < x < y < z.

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