A054875 -id:A054875 - cp's OEIS frontend

A123323 Number of integer-sided triangles with maximum side n, with sides relatively prime.

1, 1, 3, 4, 8, 7, 15, 14, 21, 20, 35, 26, 48, 39, 52, 52, 80, 57, 99, 76, 102, 95, 143, 100, 160, 132, 171, 150, 224, 148, 255, 200, 250, 224, 300, 222, 360, 279, 348, 296, 440, 294, 483, 370, 444, 407, 575, 392, 609, 460, 592, 516, 728, 495, 740, 588, 738, 644

Offset: 1

Franklin T. Adams-Watters, Sep 25 2006

Number of triples a,b,c with a <= b <= c < a+b, gcd(a,b,c) = 1 and c = n.

Dropping the requirement for side lengths to be relatively prime this sequence becomes A002620 (with a different offset). See the Sep 2006 comment in A002620. - Peter Munn, Jul 26 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Stackexchange, Number of triples for which gcd(a,b,c)=1 and c=n, Feb 25 2014

Cf. A002620, A051493, A054875, A070110, A123324, A123325, A239246.

with(numtheory):
a:= n-> add(mobius(n/d)*floor((d+1)^2/4), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 23 2013

a[n_] := DivisorSum[n, Floor[(#+1)^2/4]*MoebiusMu[n/#]&]; Array[a, 60] (* Jean-François Alcover, Dec 07 2015 *)

A123323(n)=sumdiv(n,d,floor((d+1)^2/4)*moebius(n/d))

Moebius transform of b(n) = floor((n+1)^2/4).

G.f.: (G(x)+x-x^2)/2, where G(x) = Sum_{k >= 1} mobius(k)*x^k*(1+2*x^k-x^(2*k))/(1-x^k)^2/(1-x^(2*k)).

A239246 Number of primitive Heronian triangles with n as greatest side length.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 3, 0, 0, 2, 2, 0, 0, 1, 3, 2, 0, 1, 2, 3, 0, 0, 0, 0, 2, 1, 5, 0, 4, 3, 4, 1, 0, 2, 1, 0, 0, 2, 0, 2, 4, 6, 4, 0, 2, 2, 0, 2, 0, 1, 3, 0, 1, 0, 8, 2, 0, 5, 1, 2, 0, 0, 6, 2, 7, 0, 3, 0, 0, 3, 0, 2, 0, 0, 9

Offset: 1

Frank M Jackson, Mar 13 2014

nonn

			a(17)=3 as there are 3 primitive Heronian triangles with greatest side length of 17. They are (9, 10, 17), (8, 15, 17) and (16, 17, 17).

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A054875, A070138, A096467, A123323, A134587.

nn=200; 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, c]]], {c, 3, nn}, {b, c}, {a, b}]; Table[Length@Select[lst, #==n &], {n, 1, nn}] (* using T. D. Noe's program at A083875 *)

A070787 Number of triangles with sides whose squares are integers and with positive integer area and longest side of length sqrt(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 0, 2, 1, 4, 0, 1, 3, 0, 2, 5, 5, 2, 0, 13, 0, 0, 0, 2, 9, 8, 1, 1, 9, 4, 0, 10, 0, 10, 2, 12, 11, 0, 3, 23, 14, 0, 0, 1, 13, 0, 0, 5, 5, 18, 5, 32, 18, 2, 2, 2, 0, 19, 0, 13, 16, 0, 1, 20, 35, 0, 0, 42, 0, 4, 0, 23, 24, 23, 9, 1, 0, 8, 0, 44, 10, 27, 0, 1, 48, 0, 9, 2, 27, 25, 3

Offset: 1

Henry Bottomley, May 07 2002

nonn

			a(13)=3 since the 3 triangles with sides {sqrt(13), sqrt(5), sqrt(4)}, {sqrt(13), sqrt(8), sqrt(1)} and {sqrt(13), sqrt(9), sqrt(4)} have areas 2, 1 and 3 respectively.

A. Bogomolny, Sam Loyd's Geometric Puzzle

Cf. A054875, A070783, A070784, A070785, A070786.

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

A054876 Number of pairwise incongruent triangles with integer sides and positive integer area and second longest side of length n.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 0, 1, 0, 3, 0, 2, 4, 1, 4, 1, 4, 0, 0, 6, 1, 0, 0, 3, 10, 5, 0, 4, 6, 6, 0, 1, 1, 7, 7, 2, 5, 1, 8, 10, 6, 2, 0, 2, 5, 0, 0, 3, 0, 13, 13, 14, 6, 0, 7, 5, 0, 8, 0, 14, 9, 1, 3, 1, 23, 3, 0, 13, 2, 9, 0, 6, 7, 9, 19, 4, 1, 12, 0, 14, 0, 8, 0

Offset: 1

Henry Bottomley, May 26 2000

nonn

			a(10) is 3 because there are three different integer-sided, integer-area triangles with middle side length 10, namely [9,10,17], [10,10,12], and [10,10,16].

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A054875.

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,n,z], num++], {x,1,n}], {z,n,2n}]; num]; Array[a, 100, 0] (* Amiram Eldar, Jun 19 2019 *)

Definition corrected and offset changed by James R. Buddenhagen, Jan 16 2012

More terms from Amiram Eldar, Jun 19 2019

cp's OEIS Frontend

A123323 Number of integer-sided triangles with maximum side n, with sides relatively prime.

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A239246 Number of primitive Heronian triangles with n as greatest side length.

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A070787 Number of triangles with sides whose squares are integers and with positive integer area and longest side of length sqrt(n).

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

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A054876 Number of pairwise incongruent triangles with integer sides and positive integer area and second longest side of length n.

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