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

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

Author

Amiram Eldar and Peter Munn, Nov 26 2018

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, 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

• Ray Chandler, Table of n, a(n) for n = 1..79 (terms < 6*10^6; first 67 terms from Giovanni Resta)
• Ray Chandler, First 79 terms with corresponding occurrences (first 67 terms from Giovanni Resta)

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