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 A308454 Number of integer-sided triangles with perimeter n whose largest side length is squarefree. 3
 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, 18, 17, 14, 22, 19, 18, 15, 24, 20, 19, 16, 26, 22, 33, 29, 40, 36, 35, 31, 30, 26, 38, 35, 33, 30, 28, 25, 38, 35, 48, 45, 58, 54, 51, 48, 62, 58, 73, 69, 84 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 Wikipedia, Integer Triangle FORMULA 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). EXAMPLE 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 MAPLE f:= proc(n) local p, v; v:= add(1/2*(3*p-n+1)+`if`((n-p)::even, 1/2, 0), p = select(numtheory:-issqrfree, [\$ceil(n/3)..floor((n-1)/2)])); end proc: map(f, [\$1..100]); # Robert Israel, Jan 16 2023 MATHEMATICA 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}] CROSSREFS Cf. A005044, A008683, A308453. Sequence in context: A277488 A325794 A119513 * A085815 A088234 A228717 Adjacent sequences: A308451 A308452 A308453 * A308455 A308456 A308457 KEYWORD nonn AUTHOR Wesley Ivan Hurt, May 27 2019 STATUS approved

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Last modified June 4 13:02 EDT 2023. Contains 363128 sequences. (Running on oeis4.)