A123323 -id:A123323 - cp's OEIS frontend

A051493 Triangles with perimeter n and relatively prime integer side lengths.

0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 4, 2, 5, 2, 5, 4, 8, 4, 10, 6, 9, 6, 14, 8, 15, 9, 16, 12, 21, 11, 24, 16, 22, 16, 27, 18, 33, 20, 31, 24, 40, 23, 44, 30, 39, 30, 52, 32, 54, 35, 52, 42, 65, 38, 65, 48, 64, 49, 80, 48, 85, 56, 77, 64, 90, 58, 102, 72, 93, 69, 114, 72, 120, 81

Offset: 1

Neil Fernandez

nonn

From Peter Munn, Jul 26 2017: (Start)
The triangles that meet the conditions are listed by nondecreasing n in A070110.
Without the requirement for relatively prime side lengths, this sequence becomes A005044.
Counting the triangles by longest side instead of perimeter, this sequence becomes A123323.
a(n) = A070094(n) + A070102(n) + A070109(n).
(End)

			There are 3 triangles with integer-length sides and perimeter 9: 1-4-4, 2-3-4, 3-3-3. 3-3-3 is omitted because isomorphic to 1-1-1, so a(9)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A005044, A057887, A070110, A123323.

Equivalent sequences, restricted to subsets: A070091 (isosceles), A070094 (acute), A070102 (obtuse), A070109 (right-angled), A070138 (with integer area), A070202 (with integer inradius).

nmax = 100;
A005044[n_] := Quotient[n^2 + 6n Mod[n, 2] + 24, 48];
A = Array[A005044, nmax];
mob[m_, n_] := If[ Mod[m, n] == 0, MoebiusMu[m/n], 0];
Reap[Do[Sow[Sum[mob[n, d] A[[d]], {d, 1, n}]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Oct 05 2021 *)

Moebius transform of A005044.

Corrected and extended with formula by Christian G. Bower, Nov 15 1999

Formula updated due to change to referenced sequence, and definition clarified by Peter Munn, Jul 26 2017

A316842 Three-column table read by rows giving primitive integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i, gcd(i,j,k) = 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

			Table begins:
[1,1,1],
[2,2,1],
[3,2,2],
[3,3,1],
[3,3,2],
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,3],
[5,3,3],
[5,4,2],
...

Lars Blomberg, Table of n, a(n) for n = 1..9000

There are A123323(k) rows that begin with k.
The three columns are A316846, A316847, A316848.
A316850 is a compressed version.
See A316841 for all triples (including imprimitive triples).
See A316852 and A317181 & A317183 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

A054875 Number of pairwise incongruent triangles with integer sides and positive integer area and longest side of length n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 0, 3, 1, 3, 1, 0, 3, 2, 0, 0, 3, 4, 4, 0, 1, 2, 7, 0, 1, 0, 3, 3, 2, 5, 0, 6, 7, 4, 4, 0, 2, 4, 0, 0, 5, 0, 6, 7, 10, 4, 1, 3, 4, 0, 4, 0, 10, 3, 0, 3, 1, 11, 3, 0, 8, 1, 5, 0, 4, 6, 7, 13, 0, 3, 9, 0, 10, 0, 6, 0

Offset: 1

Henry Bottomley, May 26 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A054876, A055592, A070787, A123323, A239246, 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]; Array[a, 100] (* Amiram Eldar, Nov 22 2018 *)

Name and offset changed to align with 2012 changes to A054876 by Peter Munn, Nov 23 2018

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 *)

A316846 Column 1 of table A316842.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A123323, A316842, A316847, A316848.

k appears A123323(k) times.

Terms a(54) and beyond from Lars Blomberg, Jul 27 2018

A123324 Number of integer-sided triangles with all sides <= n and sides relatively prime.

Original entry on oeis.org

1, 2, 5, 9, 17, 24, 39, 53, 74, 94, 129, 155, 203, 242, 294, 346, 426, 483, 582, 658, 760, 855, 998, 1098, 1258, 1390, 1561, 1711, 1935, 2083, 2338, 2538, 2788, 3012, 3312, 3534, 3894, 4173, 4521, 4817, 5257, 5551, 6034, 6404, 6848, 7255, 7830, 8222, 8831

Offset: 1

Franklin T. Adams-Watters, Sep 25 2006

Number of triples a,b,c with a<=b<=c

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002623, A123323, A123325.

A123323[n_] := DivisorSum[n, Floor[(#+1)^2/4]*MoebiusMu[n/#]&]; Array[ A123323, 60] // Accumulate (* Jean-François Alcover, Dec 07 2015 *)

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

Partial sums of A123323.

G.f.: (G(x)+x-x^2)/(2(1-x)), 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)).

A123325 Number of distinct angles in all integer-sided triangles with all sides <= n.

Original entry on oeis.org

1, 3, 9, 17, 35, 49, 82, 109, 149, 188, 262, 316, 419, 500, 607, 698, 876, 1004, 1222, 1383, 1589, 1782, 2108, 2318, 2634, 2914, 3253, 3564, 4088, 4411, 5000, 5392, 5917, 6410, 6995, 7468, 8308, 8926, 9661, 10268, 11313, 11976, 13136, 13951, 14875

Offset: 1

Franklin T. Adams-Watters, Sep 25 2006

nonn

Using the law of cosines, the angle A opposite side a has cos A = (b^2 + c^2 - a^2) / (2bc) and the cosine uniquely identifies an angle of a triangle.

Cf. A123323, A123324.

A123325(n)=local(i,j,k,l); r=0; l=listcreate(n^3); for(i=1,n, for(j=1,n, for(k=1,n, if(gcd(i,gcd(j,k))==1&&2*max(i,max(j,k))

A373041 2*a(n) is the number of triangles with integer sides (x, y, n), x < y < n, and gcd(x, y, n) = 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 10, 15, 15, 20, 20, 28, 24, 36, 32, 42, 40, 55, 44, 65, 57, 72, 66, 91, 68, 105, 88, 110, 100, 132, 102, 153, 126, 156, 136, 190, 138, 210, 170, 204, 187, 253, 184, 273, 215, 272, 240, 325, 234, 340, 276, 342, 301, 406, 280, 435, 345, 414, 368, 480

Offset: 5

Andrés Sancho and Hugo Pfoertner, May 21 2024

nonn

Offset 5 is chosen to exclude the only count not divisible by 2, which represents the triangle with sides (2,3,4).

David A. Corneth, Table of n, a(n) for n = 5..10000

Cf. A023022, A123323, A373051.

a(n) = {if(isprime(n), n\=2; return(n*(n-1)/2)); my(res = 0, g, sn = vecprod(factor(n)[,1])); for(b = (n + 3)\2, n-1, g = gcd(b, sn); if(g == 1, res+=(2*b - n - 1);, my(d, e); d = divisors(g); for(i = 1, #d, e = (-1)^(omega(d[i])); t = ((b-1)\d[i])*e; t-= ((n-b)\d[i])*e; res+=t))); res>>1} \\ David A. Corneth, May 22 2024

a(n) = (A373051(n) - A373051(n-1))/2 for n >= 5.

a(n) = (A123323(n) - 3*A023022(n))/2 for n >= 5.

A373051 Number of non-similar triangles possible with distinct positive integer side lengths of at most n units.

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 13, 21, 33, 47, 67, 87, 117, 147, 187, 227, 283, 331, 403, 467, 551, 631, 741, 829, 959, 1073, 1217, 1349, 1531, 1667, 1877, 2053, 2273, 2473, 2737, 2941, 3247, 3499, 3811, 4083, 4463, 4739, 5159, 5499, 5907, 6281, 6787, 7155, 7701, 8131, 8675, 9155, 9805

Offset: 1

Andrés Sancho, May 20 2024

nonn

Also, number of triangles possible with integer side lengths x, y, and z such that z < y < x <= n and gcd(x, y, z) = 1.
For all n, this number is strictly less than n^3. For all n > 5, this number is strictly greater than n.
For all n > 3, this sequence is strictly increasing.
The first n terms can be calculated in O(n^3) time.
a(n) <= A000292(n + 2). - David A. Corneth, May 22 2024

			For n = 5, the 3 solutions are (4, 3, 2), (5, 4, 2), and (5, 4, 3).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2893 terms from Andrés Sancho)
David A. Corneth, PARI program

Cf. A000292, A123323, A373041.

PARI
```
\\ See PARI link
```

a(n) = 1 + 2*Sum_{k=5..n} A373041(k) for n >= 5.

A128522 A054525 * A128174 * A000012.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 3, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 3, 2, 2, 1, 1, 2, 2, 2, 3, 2, 3, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Mar 07 2007

Row sums = A123323: (1, 1, 3, 4, 8,7, 15, 14, ...). Left column = A083290: (1, 0, 1, 1, 2, 1, 3, 2, 3, 2, ...) A128521 = A128174 * A054525 * A000012.

			First few rows of the triangle:
  1;
  0, 1;
  1, 1, 1;
  1, 1, 1, 1;
  2, 2, 2, 1, 1;
  1, 1, 1, 2, 1, 1;
  3, 3, 3, 2, 2, 1, 1;
  2, 2, 2, 2, 2, 2, 1, 1;
  3, 3, 3, 3, 3, 2, 2, 1, 1;
  ...

Cf. A054525, A128174, A000012, A083290, A123323, A128521.

A054525 * A128174 * A000012 as infinite lower triangular matrices.

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A051493 Triangles with perimeter n and relatively prime integer side lengths.

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A316842 Three-column table read by rows giving primitive integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i, gcd(i,j,k) = 1.

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A054875 Number of pairwise incongruent triangles with integer sides and positive integer area and longest side of length n.

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Mathematica

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

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A316846 Column 1 of table A316842.

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A123324 Number of integer-sided triangles with all sides <= n and sides relatively prime.

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A123325 Number of distinct angles in all integer-sided triangles with all sides <= n.

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A373041 2*a(n) is the number of triangles with integer sides (x, y, n), x < y < n, and gcd(x, y, n) = 1.

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A373051 Number of non-similar triangles possible with distinct positive integer side lengths of at most n units.

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