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A068967
Numbers that define integer Heronian triangles [a(n), prime(a(n)), A068968(n)] with area A068969(n).
3
3, 12, 25, 30, 37, 51, 60, 65, 65, 100, 116, 260, 273, 761, 761, 825, 884, 1092, 1476, 1695, 1695, 1752, 2415, 3048, 3552, 4045, 4134, 4264, 5429, 6357, 7267, 9316, 14027, 16095, 18850, 21576, 29925, 30261, 38164, 40562, 45982, 45982, 51805, 51805, 82000, 82000
OFFSET
1,1
LINKS
Eric Weisstein's World of Mathematics, Heronian Triangle.
EXAMPLE
a(5) = 37: [37, A000040(37), A068968(5)] = [37,157,130], with s = (36+157+130)/2 = 162: Area^2 = s*(s-37)*(s-157)*(s-130) = 162*125*5*32 = 3240000 = 1800*1800, therefore A068969(5) = 1800.
MATHEMATICA
area[{a_, b_, c_}] := Block[{s = (a + b + c)/2}, Sqrt[s (s-a) (s-b) (s-c)]]; zz[n_] := Block[{s, p = Prime@ n, t, z, A}, s = Solve[(n^2 - 2 n p + p^2 - z) (z - n^2 - 2 n p - p^2) == t^2 && z>0 && t>0, {t, z}, Integers]; If[s == {}, {}, z = Sort@ Select[Sqrt[z /. s], IntegerQ]]; Select[Table[{n, p, e}, {e, z}], IntegerQ[A = area[#]] && A > 0 &]]; (Join @@ Parallelize@ Array[zz, 4300])[[All, 1]] (* Giovanni Resta, Apr 20 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Mar 30 2002
EXTENSIONS
Erroneous term 2374 removed and more terms from Giovanni Resta, Apr 20 2020
STATUS
approved