login
A322105
Numbers that set a record for occurrences as longest side of a triangle with integer sides and positive integer area.
3
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
1,2
COMMENTS
Congruent triangles are identified, that is to say mirror images are not distinguished.
The corresponding numbers of occurrences are 0, 1, 2, 3, 4, 7, 10, ...
A054875(k) gives the number of occurrences for any integer k.
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)
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, 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
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar and Peter Munn, Nov 26 2018
EXTENSIONS
a(43)-a(48) from Giovanni Resta, Nov 03 2019
STATUS
approved