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A378820 a(n) is the number of distinct nondegenerate triangles whose sides are divisors of n.

%I A378820 #5 Jan 07 2025 10:21:25
%S A378820 1,3,3,6,3,11,3,10,6,10,3,26,3,10,11,15,3,23,3,23,10,10,3,46,6,10,10,
%T A378820 22,3,45,3,21,10,10,11,57,3,10,10,43,3,41,3,21,24,10,3,70,6,21,10,21,
%U A378820 3,39,10,42,10,10,3,114,3,10,23,28,10,39,3,21,10,42,3,108
%N A378820 a(n) is the number of distinct nondegenerate triangles whose sides are divisors of n.
%C A378820 A divisor can be used for several sides.
%C A378820 A nondegenerate triangle is a triangle whose sides (u, v, w) are such that u + v > w, v + w > u and u + w > v.
%H A378820 Felix Huber, <a href="/A378820/b378820.txt">Table of n, a(n) for n = 1..10000</a>
%H A378820 Wikipedia, <a href="https://en.wikipedia.org/wiki/Triangle_inequality">Triangle Inequality</a>
%F A378820 a(p) = 3 for prime p.
%e A378820 a(4) = 6 because there are the 6 distinct nondegenerate triangles (1, 1, 1), (1, 2, 2), (1, 4, 4), (2, 2, 2), (2, 4, 4), (4, 4, 4) whose sides are divisors of 4. The triples (1, 1, 2), (1, 1, 4), (1, 2, 4), (2, 2, 4) are not sides of (nondegenerate) triangles.
%p A378820 A378820:=proc(n)
%p A378820    local a,i,j,k,L;
%p A378820    L:=NumberTheory:-Divisors(n);
%p A378820    a:=0;
%p A378820    for i to nops(L) do
%p A378820       for j from i to nops(L) do
%p A378820          for k from j to nops(L) while L[k]<L[i]+L[j] do
%p A378820             a:=a+1;
%p A378820          od
%p A378820       od
%p A378820    od;
%p A378820    return a
%p A378820 end proc;
%p A378820 seq(A378820(n),n=1..72);
%Y A378820 Cf. A000005, A316841, A316842, A378819.
%K A378820 nonn
%O A378820 1,2
%A A378820 _Felix Huber_, Dec 27 2024