A308453 Number of integer-sided triangles with perimeter n whose smallest side length is squarefree.
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, 15, 12, 17, 14, 19, 16, 21, 18, 24, 20, 26, 23, 29, 25, 32, 28, 35, 31, 38, 34, 42, 37, 45, 40, 48, 42, 51, 45, 54, 47, 56, 49, 59, 52, 62, 56, 66, 59, 70, 63
Offset: 1
Keywords
Examples
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
Links
- Wikipedia, Integer Triangle
Programs
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Mathematica
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}]
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(k)^2, where mu is the Möbius function (A008683).