A396826 - OEIS

%I #5 Jun 15 2026 18:16:39

%S 0,0,0,0,0,0,1,1,4,10,15,34,53,88,147,215,339,472,715,1056,1445,2101,

%T 2871,3920,5657,7476,10318,14265,18920,25569,35052,46309,61973,83442,

%U 110887,145529,195440,255863,337317,443320,580438,758475,987242,1285233,1662587,2152775

%N 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.

%C 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).

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

%H Felix Huber, <a href="/A396826/a396826.txt">Maple program</a>

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

%e 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.

%p # See Huber link.

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

%K nonn

%O 1,9

%A _Felix Huber_, Jun 11 2026