A003057 -id:A003057 - cp's OEIS frontend

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.

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.

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.

A381563 2-tone chromatic number of a double wheel graph with n vertices.

Original entry on oeis.org

9, 9, 8, 8, 9, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15

Offset: 5

Allan Bickle, Feb 27 2025

nonn

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

A double wheel has two vertices joined to a all vertices of a cycle.

			The central vertices share exactly one color.  All vertices on the cycle require distinct pairs.
The colorings for small (broken) cycles are shown below.
  -12-34-56-
  -12-34-15-36-
  -12-34-51-23-45-
  -12-34-15-32-14-35-
  -12-34-56-13-24-35-46-
  -12-34-15-23-14-25-13-45-
  -12-34-15-32-14-25-13-24-35-

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013) 171-190.
Allan Bickle, 2-Tone Coloring of Planar Graphs, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.

Cf. A003057, A351120 (pair coloring).

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar).

a(n) = A351120(n-2) + 3 = A350715(n-1) + 1.

a(n) = ceiling((7 + sqrt(8*n - 15))/2) for n > 12.

A213194 First inverse function (numbers of rows) for pairing function A211377.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 3, 4, 4, 5, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10

Offset: 1

Boris Putievskiy, Mar 01 2013

nonn

			The start of the sequence as triangle array read by rows:
  1;
  1,1;
  2,2,3;
  1,1,2,2;
  3,3,4,4,5;
  1,1,2,2,3,3;
  4,4,5,5,6,6,7;
  1,1,2,2,3,3,4,4;
  5,5,6,6,7,7,8,8,9;
  1,1,2,2,3,3,4,4,5,5;
  . . .
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.
Last  2*r-1 numbers are from the row number 2*r-1 of above triangle array.
  1;
  1,1,2,2,3;
  1,1,2,2,3,3,4,4,5;
  1,1,2,2,3,3,4,4,5,5,6,6,7;
  1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9;
  . . .
Row r contains numbers 1,2,3,...2*r-2 repeated twice, row ends 2*r-1.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211377, A002260, A004736, A003056, A003057.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(3*i+j-1-(-1)**i+(i+j-2)*(-1)**(i+j))/4

a(n) = (3*A002600(n)+A004736(n)-1-(-1)^A002260(n)+A003056(n)*(-1)^A003057(n))/4;

a(n) = (3*i+j-1-(-1)^i+(i+j-2)*(-1)*t)/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 4, 3, 2, 12, 13, 14, 15, 11, 10, 9, 8, 7, 23, 24, 25, 26, 27, 28, 22, 21, 20, 19, 18, 17, 16, 38, 39, 40, 41, 42, 43, 44, 45, 37, 36, 35, 34, 33, 32, 31, 30, 29, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 80

Offset: 1

Boris Putievskiy, Feb 22 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(3,1), T(2,2), T(1,3);
  T(1,2), T(2,1);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(2,2*m), T(1,2*m+1);
  T(1,2*m), T(2,2*m-1), T(3,2*m-2),...T(2*m-1,2),T(2*m,1);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read downwards.

			The start of the sequence as table:
  1....5...4..12..11..23..22...
  6....3..13..10..24..21..39...
  2...14...9..25..20..40..35...
  15...8..26..19..41..34..60...
  7...27..18..42..33..61..52...
  28..17..43..32..62..51..85...
  16..44..31..63..50..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  4,3,2;
  12,13,14,15;
  11,10,9,8,7;
  23,24,25,26,27,28;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers.
If r is odd,  row is decreasing.
If r is even, row is increasing.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211394, A221215, A002260, A004736, A003057;
table T(n,k) contains: in rows A084849, A096376, A014105, A014107, A168244, A033537, A100040, A100041;
in columns A130883, A000384, A014106, A033816, A100037, A091823, A071355, A100038, A100039,A130861;
main diagonal and parallel diagonals are  A058331, A001844, A005893,A046092, A093328, A142463, A090288, A059993, A051890, A001105, A097080, A056220, A137882, A054000.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*(t+2)+4-(3*i+j-2)*(-1)**t)/2

As table
T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A003057(n)+4-(3*A002260(n)+A004736(n)-2)*(-1)^A003056(n))/2;  a(n) = ((t+2)^2-2*(t+2)+4-(i+3*j-2)*(-1)^t)/2,
  where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 6, 5, 4, 3, 2, 15, 14, 13, 12, 11, 10, 9, 8, 7, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 91

Offset: 1

Boris Putievskiy, Feb 22 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(3,1), T(2,2), T(1,3);
  T(2,1), T(1,2);
  . . .
  T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(1,2*m+1);
  T(2*m,1), T(2*m-1,2), T(2*m-2,3),...T(1,2*m);
  . . .
First row  contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)},  read upwards.

			The start of the sequence as table:
  1....6...4..15..11..28..22...
  5....3..14..10..27..21..44...
  2...13...9..26..20..43..35...
  12...8..25..19..42..34..63...
  7...24..18..41..33..62..52...
  23..17..40..32..61..51..86...
  16..39..31..60..50..85..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  6,5;
  4,3,2;
  15,14,13,12;
  11,10,9,8,7;
  28,27,26,25,24,23;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers in decreasing order.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211394, A221215, A002260, A004736, A003057, A002024;
table T(n,k) contains: in rows A084849, A000384, A014106, A014105, A014107, A091823, A071355, A091823, A071355,  A100040, A130861, A100041;
in columns A130883, A096376, A033816, A100037, A100038, A100039;
main diagonal and parallel diagonals are A058331, A051890, A005893, A097080, A093328, A137882, A001844, A001105, A056220, A142463, A054000, A090288, A059993.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*i+3-(t+1)*(1+2*(-1)**t))/2

As table
T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A002260(n)+3-A002024(n)*(1+2*(-1)^A003056(n)))/2;
a(n) = ((t+2)^2-2*i+3-(t+1)*(1+2*(-1)**t))/2, where i=n-t*(t+1)/2,
  j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

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.

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.

cp's OEIS Frontend

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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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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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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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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A381563 2-tone chromatic number of a double wheel graph with n vertices.

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A213194 First inverse function (numbers of rows) for pairing function A211377.

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A221216 T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

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A221217 T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

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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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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.

A221216 T(n,k) = ((n+k)^2-2(n+k)+4-(3n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

A221217 T(n,k) = ((n+k)^2-2n+3-(n+k-1)(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.