A349082 -id:A349082 - cp's OEIS frontend

A349090 Where zeros occur in A349082. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/x + 1/y, 0 < x < y.

10, 18, 20, 21, 28, 35, 36, 44, 45, 50, 52, 53, 54, 55, 66, 69, 70, 71, 72, 74, 75, 76, 77, 78, 84, 88, 89, 90, 91, 98, 102, 103, 104, 105, 112, 116, 118, 119, 120, 124, 125, 127, 128, 130, 131, 132, 133, 134, 135, 136, 149, 150, 152, 153, 156, 159, 160, 161

Offset: 1

Jud McCranie, Dec 12 2021

nonn

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

			10 is a term because A349082(10)=0, indicating that 4/5 = 1/x + 1/y has no solution.

Jud McCranie, Table of n, a(n) for n = 1..691

Cf. A349082, A002260, A003057, A349091, A349092.

A349095 Where ones occur in A349082. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/x + 1/y, 0 < x < y.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 25, 26, 27, 31, 32, 33, 34, 38, 40, 41, 42, 43, 46, 47, 48, 49, 51, 59, 61, 63, 64, 65, 67, 68, 73, 80, 82, 83, 85, 86, 87, 94, 96, 97, 100, 101, 110, 113, 114, 115, 117, 121, 122, 123, 126, 129, 142, 143, 144, 145, 146, 147, 148

Offset: 1

Jud McCranie, Dec 24 2021

nonn

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

			6 is a term because A349082(6)=1, indicating that 3/4 = 1/x + 1/y has a unique solution, 1/2 + 1/4.

Jud McCranie, Table of n, a(n) for n = 1..500

Cf. A349082, A002260, A003057, A349090.

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

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.

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.

Original entry on oeis.org

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.

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.

Original entry on oeis.org

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.

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

A349090 Where zeros occur in A349082. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/x + 1/y, 0 < x < y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A349095 Where ones occur in A349082. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/x + 1/y, 0 < x < y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Links

Crossrefs