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 A322105 #31 Nov 05 2023 21:27:52
%S A322105 1,5,13,15,25,30,52,65,75,100,120,145,195,300,325,390,520,585,600,650,
%T A322105 780,975,1105,1300,1560,1700,1950,2550,2600,3315,3900,4420,5100,5525,
%U A322105 6630,7800,8840,10200,11050,13260,16575,22100,26520,33150,44200,53040,66300,96135
%N A322105 Numbers that set a record for occurrences as longest side of a triangle with integer sides and positive integer area.
%C A322105 Congruent triangles are identified, that is to say mirror images are not distinguished.
%C A322105 The corresponding numbers of occurrences are 0, 1, 2, 3, 4, 7, 10, ...
%C A322105 A054875(k) gives the number of occurrences for any integer k.
%H A322105 Ray Chandler, <a href="/A322105/b322105.txt">Table of n, a(n) for n = 1..79</a> (terms < 6*10^6; first 67 terms from Giovanni Resta)
%H A322105 Ray Chandler, <a href="/A322105/a322105_1.txt">First 79 terms with corresponding occurrences</a> (first 67 terms from Giovanni Resta)
%e A322105 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 A322105 The side lengths, a(n), and their corresponding record numbers of occurrences, A054875(a(n)), are:
%e A322105 n a(n) prime factorization of a(n) occurrences
%e A322105 1 1 - 0
%e A322105 2 5 5 1
%e A322105 3 13 13 2
%e A322105 4 15 3 * 5 3
%e A322105 5 25 5^2 4
%e A322105 6 30 2 * 3 * 5 7
%e A322105 7 52 2^2 * 13 10
%e A322105 8 65 5 * 13 11
%e A322105 9 75 3 * 5^2 13
%e A322105 10 100 2^2 * 5^2 15
%e A322105 11 120 2^3 * 3 * 5 22
%e A322105 12 145 5 * 29 23
%e A322105 13 195 3 * 5 * 13 35
%e A322105 14 300 2^2 * 3 * 5^2 41
%e A322105 15 325 5^2 * 13 51
%e A322105 16 390 2 * 3 * 5 * 13 57
%e A322105 17 520 2^3 * 5 * 13 63
%e A322105 18 585 3^2 * 5 * 13 64
%e A322105 19 600 2^3 * 3 * 5^2 72
%e A322105 20 650 2 * 5^2 * 13 82
%e A322105 21 780 2^2 * 3 * 5 * 13 94
%e A322105 22 975 3 * 5^2 * 13 135
%e A322105 23 1105 5 * 13 * 17 143
%e A322105 24 1300 2^2 * 5^2 * 13 158
%e A322105 25 1560 2^3 * 3 * 5 * 13 171
%e A322105 26 1700 2^2 * 5^2 * 17 182
%e A322105 27 1950 2 * 3 * 5^2 * 13 210
%e A322105 28 2550 2 * 3 * 5^2 * 17 216
%e A322105 29 2600 2^3 * 5^2 * 13 251
%e A322105 30 3315 3 * 5 * 13 * 17 333
%e A322105 31 3900 2^2 * 3 * 5^2 * 13 367
%e A322105 32 4420 2^2 * 5 * 13 * 17 373
%e A322105 33 5100 2^2 * 3 * 5^2 * 17 406
%e A322105 34 5525 5^2 * 13 * 17 496
%e A322105 35 6630 2 * 3 * 5 * 13 * 17 525
%e A322105 36 7800 2^3 * 3 * 5^2 * 13 605
%e A322105 37 8840 2^3 * 5 * 13 * 17 610
%e A322105 38 10200 2^3 * 3 * 5^2 * 17 660
%e A322105 39 11050 2 * 5^2 * 13 * 17 735
%e A322105 40 13260 2^2 * 3 * 5 * 13 * 17 897
%e A322105 41 16575 3 * 5^2 * 13 * 17 1132
%e A322105 42 22100 2^2 * 5^2 * 13 * 17 1276
%t A322105 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
%Y A322105 Cf. A054875, A096467, A120130, A306626.
%K A322105 nonn
%O A322105 1,2
%A A322105 _Amiram Eldar_ and _Peter Munn_, Nov 26 2018
%E A322105 a(43)-a(48) from _Giovanni Resta_, Nov 03 2019