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 1-2 of 2 results.

A306626 Numbers that set a record for occurrences as longest side of a primitive Heronian triangle.

Original entry on oeis.org

1, 5, 13, 17, 37, 52, 65, 85, 119, 125, 145, 221, 325, 481, 697, 725, 1025, 1105, 1625, 1885, 2465, 2665, 3145, 5525, 6409, 15457, 15725, 26129, 27625, 38425, 40885, 45305, 58565, 67405, 69745, 83317, 128945, 160225, 204425, 226525, 237133, 292825, 348725
Offset: 1

Views

Author

Amiram Eldar and Peter Munn, Mar 01 2019

Keywords

Comments

Congruent triangles are identified, that is to say mirror images are not distinguished.
The corresponding numbers of occurrences are 0, 1, 2, 3, 5, 6, 8, ...
A239246(k) gives the number of occurrences for any integer k.
The qualifier "primitive" means that we count only triangles whose sides have a gcd of 1. The equivalent sequence without this qualification is A322105.
The terms that are common with A322105 are 1, 5, 13, 52, 65, 145, 325, 1105, 5525, ...
The odd prime factors of the terms are almost all congruent to 1 modulo 4. a(9) = 119 = 7 * 17 provides the only exception in the first 50 terms. [updated by Peter Munn, Dec 04 2019]

Examples

			13 is in the sequence since it occurs in a record number of 2 triangles of side lengths {5, 12, 13} and {10, 13, 13}.
The side lengths, a(n), and their corresponding record numbers of occurrences, A239246(a(n)), are:
n       a(n)     prime factorization of a(n)  occurrences
1          1     -                               0
2          5     5                               1
3         13     13                              2
4         17     17                              3
5         37     37                              5
6         52     2^2 * 13                        6
7         65     5 * 13                          8
8         85     5 * 17                          9
9        119     7 * 17                         10
10       125     5^3                            13
11       145     5 * 29                         20
12       221     13 * 17                        30
13       325     5^2 * 13                       37
14       481     13 * 37                        42
15       697     17 * 41                        50
16       725     5^2 * 29                       54
17      1025     5^2 * 41                       63
18      1105     5 * 13 * 17                    90
19      1625     5^3 * 13                       93
20      1885     5 * 13 * 29                   106
21      2465     5 * 17 * 29                   116
22      2665     5 * 13 * 41                   134
23      3145     5 * 17 * 37                   178
24      5525     5^2 * 13 * 17                 277
25      6409     13 * 17 * 29                  373
26     15457     13 * 29 * 41                  396
27     15725     5^2 * 17 * 37                 463

Links

Crossrefs

Cf. A096467, A120130, A239246, A322105.

Programs

  • Mathematica
    okQ[x_, y_, z_] := GCD[x, y, z]==1 && If[x + y <= z, False, Module[{s = (x + y + z)/2}, IntegerQ[ Sqrt[s(s-x)(s-y)(s-z)]]] ]; a[n_] := Module[{num = 0}, Do[Do[If[okQ[x, y, n], num++], {x, 1, y}], {y, 1, n}]; num]; amax=-1; s={}; Do[a1=a[n]; If[a1 > amax, AppendTo[s, n]; amax=a1], {n, 1, 100}]; s

Extensions

a(28)-a(43) from Giovanni Resta, Nov 07 2019

A322105 Numbers that set a record for occurrences as longest side of a triangle with integer sides and positive integer area.

Original entry on oeis.org

1, 5, 13, 15, 25, 30, 52, 65, 75, 100, 120, 145, 195, 300, 325, 390, 520, 585, 600, 650, 780, 975, 1105, 1300, 1560, 1700, 1950, 2550, 2600, 3315, 3900, 4420, 5100, 5525, 6630, 7800, 8840, 10200, 11050, 13260, 16575, 22100, 26520, 33150, 44200, 53040, 66300, 96135
Offset: 1

Views

Author

Amiram Eldar and Peter Munn, Nov 26 2018

Keywords

Comments

Congruent triangles are identified, that is to say mirror images are not distinguished.
The corresponding numbers of occurrences are 0, 1, 2, 3, 4, 7, 10, ...
A054875(k) gives the number of occurrences for any integer k.

Examples

			13 is in the sequence since it occurs in a record number of 2 triangles of side lengths {5, 12, 13} and {10, 13, 13}.
The side lengths, a(n), and their corresponding record numbers of occurrences, A054875(a(n)), are:
n       a(n)     prime factorization of a(n)  occurrences
1          1     -                               0
2          5     5                               1
3         13     13                              2
4         15     3 * 5                           3
5         25     5^2                             4
6         30     2 * 3 * 5                       7
7         52     2^2 * 13                       10
8         65     5 * 13                         11
9         75     3 * 5^2                        13
10       100     2^2 * 5^2                      15
11       120     2^3 * 3 * 5                    22
12       145     5 * 29                         23
13       195     3 * 5 * 13                     35
14       300     2^2 * 3 * 5^2                  41
15       325     5^2 * 13                       51
16       390     2 * 3 * 5 * 13                 57
17       520     2^3 * 5 * 13                   63
18       585     3^2 * 5 * 13                   64
19       600     2^3 * 3 * 5^2                  72
20       650     2 * 5^2 * 13                   82
21       780     2^2 * 3 * 5 * 13               94
22       975     3 * 5^2 * 13                  135
23      1105     5 * 13 * 17                   143
24      1300     2^2 * 5^2 * 13                158
25      1560     2^3 * 3 * 5 * 13              171
26      1700     2^2 * 5^2 * 17                182
27      1950     2 * 3 * 5^2 * 13              210
28      2550     2 * 3 * 5^2 * 17              216
29      2600     2^3 * 5^2 * 13                251
30      3315     3 * 5 * 13 * 17               333
31      3900     2^2 * 3 * 5^2 * 13            367
32      4420     2^2 * 5 * 13 * 17             373
33      5100     2^2 * 3 * 5^2 * 17            406
34      5525     5^2 * 13 * 17                 496
35      6630     2 * 3 * 5 * 13 * 17           525
36      7800     2^3 * 3 * 5^2 * 13            605
37      8840     2^3 * 5 * 13 * 17             610
38     10200     2^3 * 3 * 5^2 * 17            660
39     11050     2 * 5^2 * 13 * 17             735
40     13260     2^2 * 3 * 5 * 13 * 17         897
41     16575     3 * 5^2 * 13 * 17            1132
42     22100     2^2 * 5^2 * 13 * 17          1276

Links

Crossrefs

Cf. A054875, A096467, A120130, A306626.

Programs

  • Mathematica
    okQ[x_, y_, z_] := If[x + y <= z, False, Module[{s = (x + y + z)/2}, IntegerQ[ Sqrt[s(s-x)(s-y)(s-z)]]] ]; a[n_] := Module[{num = 0}, Do[Do[If[okQ[x, y, n], num++], {x, 1, y}], {y, 1, n}]; num]; amax=-1; s={}; Do[a1=a[n]; If[a1 > amax, AppendTo[s, n]; amax=a1],{n,1,100}]; s

Extensions

a(43)-a(48) from Giovanni Resta, Nov 03 2019
Showing 1-2 of 2 results.

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