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

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

Author

Amiram Eldar and Peter Munn, Mar 01 2019

Keywords

• nonn

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

• Ray Chandler, Table of n, a(n) for n = 1..67 (terms < 6*10^6; first 50 terms from Giovanni Resta)

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