cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 106 results. Next

A358886 Number of regions formed inside a square with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Original entry on oeis.org

4, 56, 1040, 6064, 53104, 115496, 629920, 1457744, 3952264, 6835568
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).
We call this graph the Completed Farey Diagram of order (n,k). The (ordinary) Farey diagram Farey(n,k) is a subgraph. In the latter graph, not all pairs of boundary points are joined by lines.

Links

Crossrefs

Cf. A358888 (edges), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882.
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Formula

a(n) = A358888(n) - A358887(n) + 1 by Euler's formula.

A358889 Table read by rows: T(n,k) = number of k-gons, k >= 3, formed inside a square with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,m)/A006843(n,m), m = 1..A005728(n).

Original entry on oeis.org

4, 48, 8, 712, 304, 24, 3368, 2400, 280, 16, 27424, 20360, 4784, 504, 32, 56000, 47088, 10912, 1400, 88, 8, 292424, 255608, 69368, 11504, 960, 56, 658800, 590208, 175856, 30160, 2496, 200, 24, 1748112, 1593912, 506496, 93584, 9616, 520, 24, 2981448, 2778456, 890368, 166912, 17192, 1144, 48
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).

Examples

			The table begins:
 4;
 48,      8;
 712,     304,     24;
 3368,    2400,    280,    16;
 27424,   20360,   4784,   504,    32;
 56000,   47088,   10912,  1400,   88,    8;
 292424,  255608,  69368,  11504,  960,   56;
 658800,  590208,  175856, 30160,  2496,  200,  24;
 1748112, 1593912, 506496, 93584,  9616,  520,  24;
 2981448, 2778456, 890368, 166912, 17192, 1144, 48;
.
.

Links

Crossrefs

Cf. A358886 (regions), A358887 (vertices), A358888 (edges), A006842, A006843, A005728, A358885.
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Formula

Sum of row n = A358886(n).

A358887 Number of vertices formed inside a square with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Original entry on oeis.org

5, 37, 705, 4549, 42357, 94525, 531485, 1250681, 3440621, 5985201
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).

Links

Crossrefs

Cf. A358888 (edges), A358886 (regions), A358889 (k-gons), A006842, A006843, A005728, A358882, A358883.
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Formula

a(n) = A358888(n) - A358886(n) + 1 by Euler's formula.

A358888 Number of edges formed inside a square with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Original entry on oeis.org

8, 92, 1744, 10612, 95460, 210020, 1161404, 2708424, 7392884, 12820768
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).
See A358886 and A358887 for images of the square.

Links

Crossrefs

Cf. A358886 (regions), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882, A358884.
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Formula

a(n) = A358886(n) + A358887(n) - 1 by Euler's formula.

A358948 Number of regions formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Original entry on oeis.org

1, 12, 228, 1464, 12516, 29022, 153564, 364650, 996672, 1750326, 5274156, 7761498
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).

Links

Crossrefs

Cf. A358949 (vertices), A358950 (edges), A358951 (k-gons), A358886, A006842, A006843, A005728, A358882.

Formula

a(n) = A358950(n) - A358949(n) + 1 by Euler's formula.

A358949 Number of vertices formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Original entry on oeis.org

3, 10, 148, 1111, 9568, 23770, 126187, 308401, 855145, 1521733, 4591405, 6831040
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).

Links

Crossrefs

Cf. A358948 (regions), A358950 (edges), A358951 (k-gons), A358887, A006842, A006843, A005728, A358882.

Formula

a(n) = A358950(n) - A358948(n) + 1 by Euler's formula.

A358951 Irregular table read by rows: T(n,k) = number of k-gons, k >= 3, formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,m)/A006843(n,m), m = 1..A005728(n).

Original entry on oeis.org

1, 12, 180, 42, 6, 810, 576, 72, 6, 6786, 4932, 744, 48, 6, 13662, 12522, 2568, 258, 12, 72582, 64932, 14376, 1632, 36, 6, 164484, 155088, 38688, 5958, 414, 18, 439524, 422370, 114804, 18462, 1392, 120, 750108, 749928, 211518, 35226, 3336, 204, 6, 2265462, 2240994, 647184, 109602, 10230, 666, 18
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).

Examples

			The table begins:
1;
12;
180, 42, 6;
810, 576, 72, 6;
6786, 4932, 744, 48, 6;
13662, 12522, 2568, 258, 12;
72582, 64932, 14376, 1632, 36, 6;
164484, 155088, 38688, 5958, 414, 18;
439524, 422370, 114804, 18462, 1392, 120;
750108, 749928, 211518, 35226, 3336, 204, 6;
2265462, 2240994, 647184, 109602, 10230, 666, 18;
3263436, 3312270, 990072, 176172, 18294, 1188, 66;
.
.

Links

Crossrefs

Cf. A358948 (regions), A358949 (vertices), A358950 (edges), A358889, A006842, A006843, A005728, A358882.

Formula

Sum of row n = A358948(n).

A358950 Number of edges formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Original entry on oeis.org

3, 21, 375, 2574, 22083, 52791, 279750, 673050, 1851816, 3272058, 9865560, 14592537
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).
See A358948 and A358949 for images of the square.

Links

Crossrefs

Cf. A358948 (regions), A358949 (vertices), A358951 (k-gons), A358888, A006842, A006843, A005728, A358882.

Formula

a(n) = A358948(n) + A358949(n) - 1 by Euler's formula.

A055197 Numbers k such that A005728(k) is not prime.

Original entry on oeis.org

10, 14, 16, 19, 20, 21, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 56, 57, 58, 64, 65, 66, 67, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 89, 90, 91, 92, 94, 95, 96, 98, 99, 100
Offset: 1

Views

Author

Robert G. Wilson v, Jul 04 2000

Keywords

Comments

From a question posed by Leo Moser.

References

  • Martin Gardner, "The Last Recreations," Chapter entitled "Strong Laws of Small Primes," Copernicus, Springer-Verlag, NY, 1997, page 199.

Links

Crossrefs

Cf. A005728.
Complement of A055201.

Programs

  • Mathematica
    s=1; Do[ s=s+EulerPhi[ n ]; If[ !PrimeQ[ s ], Print[ n ] ], {n, 1, 100} ]

Extensions

Offset corrected by Amiram Eldar, Mar 01 2020

A055201 Numbers k such that A005728(k) is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 17, 18, 22, 23, 24, 29, 36, 37, 40, 53, 55, 59, 60, 61, 62, 63, 68, 69, 74, 83, 86, 93, 97, 107, 109, 115, 116, 117, 118, 123, 128, 129, 139, 142, 147, 153, 154, 157, 158, 162, 167, 170, 175, 179, 181, 182, 183, 193, 194, 196
Offset: 1

Views

Author

Robert G. Wilson v, Jul 04 2000

Keywords

Comments

From a question by Leo Moser.

References

  • Martin Gardner, "The Last Recreations," Chapter entitled "Strong Laws of Small Primes," Copernicus, Springer-Verlag, NY, 1997, page 199.

Links

Crossrefs

Cf. A005728.
Complement of A055197.

Programs

  • Mathematica
    s=1; Do[s = s + EulerPhi[n]; If[PrimeQ[s], Print[n]], {n, 200}]
Showing 1-10 of 106 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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