A396826 a(n) is the number of distinct integer-sided triangles (x, y, z) with sopfr(x) = sopfr(y) = sopfr(z) = n, where sopfr is A001414.

0, 0, 0, 0, 0, 0, 1, 1, 4, 10, 15, 34, 53, 88, 147, 215, 339, 472, 715, 1056, 1445, 2101, 2871, 3920, 5657, 7476, 10318, 14265, 18920, 25569, 35052, 46309, 61973, 83442, 110887, 145529, 195440, 255863, 337317, 443320, 580438, 758475, 987242, 1285233, 1662587, 2152775

Offset

1,9

Comments

Let S(n) be the set of positive integers m with sopfr(m) = n; then |S(n)| = A000607(n), and a(n) counts the triangles formed by elements of S(n).

The first Heronian example occurs for n = 54: (629, 13195, 13320).

Links

Table of n, a(n) for n=1..46.

Felix Huber, Maple program

Formula

A396827(n) <= a(n) <= binomial(A000607(n), 3).

Example

a(9) = 4 since the numbers with sopfr equal to 9 are 14, 18, 20 and 27, and the triangles formed from them are (14, 18, 20), (14, 18, 27), (14, 20, 27) and (18, 20, 27); among these, only (14, 18, 20) is not primitive.

Maple

# See Huber link.

Crossrefs

Cf. A000607, A001414, A005044, A024162, A046675, A396827.

Sequence in context: A360557 A005662 A337294 * A259262 A218211 A302197

Adjacent sequences: A396821 A396822 A396823 * A396827 A396828 A396829

Keyword

nonn

Author

Felix Huber, Jun 11 2026

Status

approved