OFFSET
1,2
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]
LINKS
Ray Chandler, Table of n, a(n) for n = 1..67 (terms < 6*10^6; first 50 terms from Giovanni Resta)
EXAMPLE
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
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar and Peter Munn, Mar 01 2019
EXTENSIONS
a(28)-a(43) from Giovanni Resta, Nov 07 2019
STATUS
approved