cp's OEIS Frontend

A333917 Perimeters of integer-sided triangles whose altitude from their longest side is an integer.

%I A333917 #15 Feb 16 2025 08:33:59
%S A333917 16,18,32,36,48,50,54,60,64,70,72,80,90,96,98,100,108,112,120,126,128,
%T A333917 132,140,144,150,154,160,162,168,176,180,192,196,198,200,208,210,216,
%U A333917 220,224,234,240,242,250,252,256,260,264,270,272,280,286,288,290,294,300
%N A333917 Perimeters of integer-sided triangles whose altitude from their longest side is an integer.
%H A333917 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Altitude.html">Altitude</a>
%H A333917 Wikipedia, <a href="https://en.wikipedia.org/wiki/Altitude_(triangle)">Altitude (triangle)</a>
%H A333917 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%e A333917 16 is in the sequence since it is the perimeter of the triangle [5,5,6], whose altitude from 6 (the longest side) is 4 (an integer).
%e A333917 18 is in the sequence since it is the perimeter of the triangle [5,5,8], whose altitude from 8 (the longest side) is 3 (an integer).
%e A333917 48 is in the sequence since it is the perimeter of the triangles [15,15,18] and [10,17,21], whose altitudes from their longest sides are 12 and 8 respectively (both integers).
%t A333917 Flatten[Table[If[Sum[Sum[(1 - Ceiling[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/(n - i - k)] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/(n - i - k)]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}] > 0, n, {}], {n, 100}]]
%Y A333917 Cf. A005044, A333918, A333919.
%K A333917 nonn
%O A333917 1,1
%A A333917 _Wesley Ivan Hurt_, Apr 09 2020