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A308454 Number of integer-sided triangles with perimeter n whose largest side length is squarefree.

%I A308454 #15 Jan 24 2023 13:52:26
%S A308454 0,0,1,0,1,1,2,1,1,0,3,2,5,4,7,5,4,3,2,1,6,5,10,9,8,7,13,11,17,15,21,
%T A308454 18,17,14,22,19,18,15,24,20,19,16,26,22,33,29,40,36,35,31,30,26,38,35,
%U A308454 33,30,28,25,38,35,48,45,58,54,51,48,62,58,73,69,84
%N A308454 Number of integer-sided triangles with perimeter n whose largest side length is squarefree.
%H A308454 Robert Israel, <a href="/A308454/b308454.txt">Table of n, a(n) for n = 1..10000</a>
%H A308454 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F A308454 a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(n-i-k)^2, where mu is the Möbius function (A008683).
%e A308454 There exist A005044(11) = 4 integer-sided triangles with perimeter = 11; these four triangles have respectively sides: (1, 5, 5); (2, 4, 5); (3, 3, 5); (3, 4, 4). Only the last one: (3, 4, 4) has a largest side length = 4 that is not squarefree, so a(11) = 3. - _Bernard Schott_, Jan 24 2023
%p A308454 f:= proc(n) local p, v;
%p A308454   v:= add(1/2*(3*p-n+1)+`if`((n-p)::even, 1/2, 0),
%p A308454      p = select(numtheory:-issqrfree, [$ceil(n/3)..floor((n-1)/2)]));
%p A308454 end proc:
%p A308454 map(f, [$1..100]); # _Robert Israel_, Jan 16 2023
%t A308454 Table[Sum[Sum[ MoebiusMu[n - i - k]^2* Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y A308454 Cf. A005044, A008683, A308453.
%K A308454 nonn
%O A308454 1,7
%A A308454 _Wesley Ivan Hurt_, May 27 2019