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A333918
Perimeters of integer-sided triangles whose altitude from their shortest side is an integer.
3
12, 24, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 80, 84, 88, 90, 96, 98, 100, 104, 108, 112, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196, 198, 200
OFFSET
1,1
LINKS
Eric Weisstein's World of Mathematics, Altitude
Wikipedia, Integer Triangle
FORMULA
12 is in the sequence since it is the perimeter of the triangle [3,4,5], whose altitude from 3 (the shortest side) is 4 (an integer).
24 is in the sequence since it is the perimeter of the triangle [6,8,10], whose altitude from 6 (the shortest side) is 8 (an integer).
54 is in the sequence since it is the perimeter of the triangles [3,25,26] and [12,17,25] whose altitudes from their shortest sides are 24 and 15 respectively (both integers).
MATHEMATICA
Flatten[Table[If[Sum[Sum[(1 - Ceiling[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/k] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}] > 0, n, {}], {n, 100}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Apr 09 2020
STATUS
approved