Andrés Sancho - cp's OEIS frontend

User: Andrés Sancho

Andrés Sancho has authored 2 sequences.

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

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.

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.

cp's OEIS Frontend

User: Andrés Sancho

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula