A308453 - OEIS

%I #14 Jan 24 2023 02:47:29

%S 0,0,1,0,1,1,2,1,3,2,4,2,4,2,5,3,6,5,8,6,10,8,12,9,13,10,14,10,14,11,

%T 15,12,17,14,19,16,21,18,24,20,26,23,29,25,32,28,35,31,38,34,42,37,45,

%U 40,48,42,51,45,54,47,56,49,59,52,62,56,66,59,70,63

%N Number of integer-sided triangles with perimeter n whose smallest side length is squarefree.

%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))) * mu(k)^2, where mu is the Möbius function (A008683).

%e There exist A005044(12) = 3 integer-sided triangles with perimeter = 12; these three triangles have respectively sides: (2, 5, 5), (3, 4, 5) or (4, 4, 4). Only the last one: (4, 4, 4) has a smallest side length = 4 that is not squarefree, so a(12) = 2. - _Bernard Schott_, Jan 22 2023

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

%Y Cf. A005044, A008683, A308454.

%K nonn

%O 1,7

%A _Wesley Ivan Hurt_, May 27 2019