A306626 -id:A306626 - cp's OEIS frontend

A120131 Longest side of primitive Heronian triangles, sorted.

5, 6, 8, 13, 13, 15, 15, 17, 17, 17, 20, 20, 21, 21, 24, 25, 25, 25, 26, 26, 28, 29, 29, 30, 30, 30, 35, 35, 36, 37, 37, 37, 37, 37, 39, 39, 39, 39, 40, 40, 40, 41, 41, 41, 41, 42, 44, 44, 45, 48, 48, 50, 50, 51, 51, 51, 51, 52, 52, 52, 52, 52, 52, 53, 53, 53, 53, 55, 55, 56

Offset: 1

Lekraj Beedassy, Jun 10 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000
Michael Somos, Heronian Triangle Table
P. Yiu, Heron triangles with sides < 100, Recreational Mathematics, Appendix Chap. 9.3 pp. 81/360. (This is a download of 360 pages.)

Cf. A055592, A120132, A120133, A072294, A306626.

hQ[a_,b_,c_] := IntegerQ@ Sqrt@ Block[{s = (a+b+c)/2}, s (s-a) (s-b) (s-c)]; Reap[Do[If[ GCD[a, b, c] == 1 && hQ[a, b, c], Sow@ a], {a, 60}, {b, a}, {c, a-b+1, b}]][[2, 1]] (* Giovanni Resta, May 21 2016 *)

A096467 Numbers that can be the longest side of a primitive Heronian triangle.

Original entry on oeis.org

5, 6, 8, 13, 15, 17, 20, 21, 24, 25, 26, 28, 29, 30, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 69, 70, 73, 74, 75, 77, 80, 82, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113

Offset: 1

T. D. Noe, Jun 22 2004

nonn

Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area. Note that all primes of the form 4k+1 are in this sequence. It appears that a prime of the form 4k+3 is never the longest side of a Heronian triangle. Cheney's article contains many theorems about these triangles.

			5 is on this list because the triangle with sides 3, 4, 5 has integral area.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 240 terms from Vincenzo Librandi)
Wm. Fitch Cheney, Jr., Heronian Triangles, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 22-28.
Eric Weisstein's World of Mathematics, Heronian Triangle

Cf. A083875 (area/6 of primitive Heronian triangles), A096468 (perimeter of primitive Heronian triangles).

Cf. A239246, A306626.

nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, a]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

A120130 First value for longest side common to exactly n primitive Heronian triangles.

Original entry on oeis.org

5, 13, 17, 39, 37, 52, 75, 65, 85, 119, 185, 229, 125, 169, 205, 241, 409, 195, 615, 145, 265, 445, 1111, 340, 555, 507, 663, 485, 765, 221, 493, 377, 715, 865, 689, 425, 325, 1465, 1131, 845, 1015, 481, 3365, 2165, 1037, 949, 901, 1855, 4715, 697, 2516, 925

Offset: 1

Lekraj Beedassy, Jun 10 2006

nonn

Ray Chandler, Table of n, a(n) for n = 1..2819 (first 1247 terms from Giovanni Resta)
Michael Somos, Heronian Triangle Table
P. Yiu, Heron triangles with sides < 100, Recreational Mathematics, Appendix Chap. 9.3 pp. 81/360. (This is a download of 360 pages.)

Cf. A120131, A306626.

More terms from Giovanni Resta, Nov 07 2019

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

Original entry on oeis.org

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

Amiram Eldar and Peter Munn, Nov 26 2018

nonn

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.

			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

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)

Cf. A054875, A096467, A120130, A306626.

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

a(43)-a(48) from Giovanni Resta, Nov 03 2019

cp's OEIS Frontend

A120131 Longest side of primitive Heronian triangles, sorted.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A096467 Numbers that can be the longest side of a primitive Heronian triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A120130 First value for longest side common to exactly n primitive Heronian triangles.

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions