A337086 - OEIS

%I #14 Feb 01 2021 21:14:28

%S 0,0,1,0,0,1,0,0,2,1,0,1,1,0,1,0,0,2,0,1,1,0,0,1,0,1,2,1,0,2,1,0,1,0,

%T 1,3,1,0,2,1,0,2,0,0,4,1,1,2,0,1,1,2,1,2,1,2,1,0,0,2,1,2,3,1,1,1,0,0,

%U 1,2,1,3,1,1,1,0,1,2,0,1,2,0,0,3,3,1,1,0,2,6,3,1,2,1,0,2

%N Number of integer-sided triangles with perimeter n where the harmonic mean of the side lengths is an integer.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>

%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * chi(3*i*k*(n-i-k)/(i*k+k*(n-i-k)+i*(n-i-k))), where chi(n) = 1 - ceiling(n) + floor(n).

%F a(n) = A005044(n) - A337087(n).

%e a(6) = 1; There is one integer-sided triangle with perimeter 6, [2,2,2]. The harmonic mean of its side lengths is 3*2*2*2/(2*2+2*2+2*2) = 2 (an integer).

%e a(10) = 1; There are 2 integer-sided triangles with perimeter 10, [2,4,4] and [3,3,4]. For the harmonic mean of the [2,4,4] triangle, we get 3*2*4*4/(2*4+2*4+4*4) = 96/32 = 3 (an integer), but the harmonic mean for the [3,3,4] triangle is 3*3*3*4/(3*3+3*4+3*4) = 108/33 (not an integer). Thus, a(10) = 1.

%t Table[Sum[Sum[(1 - Ceiling[3*i*k*(n - i - k)/(i*k + k*(n - i - k) + i*(n - i - k))] + Floor[3*i*k*(n - i - k)/(i*k + k*(n - i - k) + i*(n - i - k))]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

%Y Cf. A005044, A337087.

%K nonn,easy

%O 1,9

%A _Wesley Ivan Hurt_, Aug 14 2020