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A378819 a(n) is the number of distinct nondegenerate triangles whose sides are prime factors of n.

%I A378819 #6 Jan 07 2025 10:21:16
%S A378819 0,1,1,1,1,4,1,1,1,3,1,4,1,3,4,1,1,4,1,3,3,3,1,4,1,3,1,3,1,8,1,1,3,3,
%T A378819 4,4,1,3,3,3,1,7,1,3,4,3,1,4,1,3,3,3,1,4,3,3,3,3,1,8,1,3,3,1,3,7,1,3,
%U A378819 3,7,1,4,1,3,4,3,4,7,1,3,1,3,1,7,3,3,3,3
%N A378819 a(n) is the number of distinct nondegenerate triangles whose sides are prime factors of n.
%C A378819 A prime factor can be used for several sides.
%C A378819 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 A378819 Felix Huber, <a href="/A378819/b378819.txt">Table of n, a(n) for n = 1..10000</a>
%H A378819 Wikipedia, <a href="https://en.wikipedia.org/wiki/Triangle_inequality">Triangle Inequality</a>
%F A378819 a(n) = a(A007947(n)).
%F A378819 a(p^k) = 1 for prime powers p^k (p prime, k >= 1).
%e A378819 a(10) = 3 because there are the 3 distinct nondegenerate triangles (2, 2, 2), (2, 5, 5), (5, 5, 5) whose sides are prime factors of 10. Since 2 + 2 < 5, (2, 2, 5) is not a triangle.
%p A378819 A378819:=proc(n)
%p A378819    local a,i,j,k,L;
%p A378819    L:=NumberTheory:-PrimeFactors(n);
%p A378819    a:=0;
%p A378819    for i to nops(L) do
%p A378819       for j from i to nops(L) do
%p A378819          for k from j to nops(L) while L[k]<L[i]+L[j] do
%p A378819             a:=a+1;
%p A378819          od
%p A378819       od
%p A378819    od;
%p A378819    return a
%p A378819 end proc;
%p A378819 seq(A378819(n),n=1..88);
%Y A378819 Cf. A000040, A000961, A001221, A007947, A070088, A306678, A316841, A316842, A366398, A378820, A379033.
%K A378819 nonn
%O A378819 1,6
%A A378819 _Felix Huber_, Dec 27 2024