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.
%I A306626 #25 Nov 05 2023 15:22:54
%S A306626 1,5,13,17,37,52,65,85,119,125,145,221,325,481,697,725,1025,1105,1625,
%T A306626 1885,2465,2665,3145,5525,6409,15457,15725,26129,27625,38425,40885,
%U A306626 45305,58565,67405,69745,83317,128945,160225,204425,226525,237133,292825,348725
%N A306626 Numbers that set a record for occurrences as longest side of a primitive Heronian triangle.
%C A306626 Congruent triangles are identified, that is to say mirror images are not distinguished.
%C A306626 The corresponding numbers of occurrences are 0, 1, 2, 3, 5, 6, 8, ...
%C A306626 A239246(k) gives the number of occurrences for any integer k.
%C A306626 The qualifier "primitive" means that we count only triangles whose sides have a gcd of 1. The equivalent sequence without this qualification is A322105.
%C A306626 The terms that are common with A322105 are 1, 5, 13, 52, 65, 145, 325, 1105, 5525, ...
%C A306626 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]
%H A306626 Ray Chandler, <a href="/A306626/b306626.txt">Table of n, a(n) for n = 1..67</a> (terms < 6*10^6; first 50 terms from Giovanni Resta)
%e A306626 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}.
%e A306626 The side lengths, a(n), and their corresponding record numbers of occurrences, A239246(a(n)), are:
%e A306626 n a(n) prime factorization of a(n) occurrences
%e A306626 1 1 - 0
%e A306626 2 5 5 1
%e A306626 3 13 13 2
%e A306626 4 17 17 3
%e A306626 5 37 37 5
%e A306626 6 52 2^2 * 13 6
%e A306626 7 65 5 * 13 8
%e A306626 8 85 5 * 17 9
%e A306626 9 119 7 * 17 10
%e A306626 10 125 5^3 13
%e A306626 11 145 5 * 29 20
%e A306626 12 221 13 * 17 30
%e A306626 13 325 5^2 * 13 37
%e A306626 14 481 13 * 37 42
%e A306626 15 697 17 * 41 50
%e A306626 16 725 5^2 * 29 54
%e A306626 17 1025 5^2 * 41 63
%e A306626 18 1105 5 * 13 * 17 90
%e A306626 19 1625 5^3 * 13 93
%e A306626 20 1885 5 * 13 * 29 106
%e A306626 21 2465 5 * 17 * 29 116
%e A306626 22 2665 5 * 13 * 41 134
%e A306626 23 3145 5 * 17 * 37 178
%e A306626 24 5525 5^2 * 13 * 17 277
%e A306626 25 6409 13 * 17 * 29 373
%e A306626 26 15457 13 * 29 * 41 396
%e A306626 27 15725 5^2 * 17 * 37 463
%t A306626 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
%Y A306626 Cf. A096467, A120130, A239246, A322105.
%K A306626 nonn
%O A306626 1,2
%A A306626 _Amiram Eldar_ and _Peter Munn_, Mar 01 2019
%E A306626 a(28)-a(43) from _Giovanni Resta_, Nov 07 2019