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

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, 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, 3, 7, 1, 4, 1, 3, 4, 3, 4, 7, 1, 3, 1, 3, 1, 7, 3, 3, 3, 3

Offset

1,6

Comments

A prime factor can be used for several sides.

A nondegenerate triangle is a triangle whose sides (u, v, w) are such that u + v > w, v + w > u and u + w > v.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Wikipedia, Triangle Inequality

Formula

a(n) = a(A007947(n)).

a(p^k) = 1 for prime powers p^k (p prime, k >= 1).

Example

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.

Maple

A378819:=proc(n)
   local a, i, j, k, L;
   L:=NumberTheory:-PrimeFactors(n);
   a:=0;
   for i to nops(L) do
      for j from i to nops(L) do
         for k from j to nops(L) while L[k]<L[i]+L[j] do
            a:=a+1;
         od
      od
   od;
   return a
end proc;
seq(A378819(n), n=1..88);

Crossrefs

Cf. A000040, A000961, A001221, A007947, A070088, A306678, A316841, A316842, A366398, A378820, A379033.

Sequence in context: A321592 A031278 A010328 * A332847 A245501 A247026

Adjacent sequences: A378816 A378817 A378818 * A378820 A378821 A378822

Keyword

nonn

Author

Felix Huber, Dec 27 2024

Status

approved