A239978 -id:A239978 - cp's OEIS frontend

A227166 Areas of indecomposable non-Pythagorean primitive integer Heronian triangles, sorted increasingly.

72, 126, 168, 252, 252, 288, 336, 336, 396, 396, 420, 420, 420, 420, 456, 462, 528, 528, 624, 714, 720, 720, 756, 792, 798, 840, 840, 840, 840, 864, 924, 924, 924, 924, 936, 990, 1008, 1092, 1092, 1188, 1200, 1218, 1248, 1260, 1260, 1320, 1320, 1320

Offset: 1

Frank M Jackson, Jul 03 2013

nonn

An indecomposable integer Heronian triangle that is not Pythagorean cannot be decomposed into two separate Pythagorean triangles because it has no integer altitudes.

See comments in A227003 about the Mathematica program below to ensure that all primitive Heronian areas up to 1320 are captured.

			a(2) = 126 as this is the second smallest area of an indecomposable non-Pythagorean primitive Heronian triangle. The triple is (5,51,52).

Paul Yiu, Heron triangles which cannot be decomposed into two integer right triangles, 2008

Cf. A083875, A224301, A227003, A239978.

nn=1320; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s]&&GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]] && !IntegerQ[2Sqrt[area2]/a] && !IntegerQ[2Sqrt[area2]/b] && !IntegerQ[2Sqrt[area2]/c], AppendTo[lst, Sqrt[area2]]]], {a, 3, nn}, {b, a}, {c, b}]; Sort@Select[lst, #<=nn &] (* using T. D. Noe's program A083875 *)

Name clarified by Frank M Jackson, Mar 17 2014

A240240 Consider primitive Heronian triangles with integer area and with sides {m, m+1, c}, where c > m+1. The sequence gives the possible values of m.

Original entry on oeis.org

3, 9, 13, 19, 20, 33, 51, 65, 73, 99, 119, 129, 163, 170, 174, 193, 201, 203, 220, 243, 260, 269, 287, 289, 339, 362, 377, 393, 450, 451, 513, 532, 559, 579, 615, 649, 696, 702, 714, 723, 740, 771, 801, 883, 909, 940, 969, 975, 1059, 1112, 1153, 1155, 1156, 1164, 1251, 1299, 1325, 1332, 1353, 1424, 1455, 1459, 1569, 1605, 1615, 1683, 1690, 1716, 1801, 1869, 1919, 1923

Offset: 1

Zak Seidov, Apr 03 2014

nonn

Corresponding values of c are 5, 17, 15, 37, 29, 65, 101, 109, 145.
And corresponding values of area/6 are 1, 6, 14, 19, 35, 44, 85, 330, 146, 231, 1190.
The sequence includes all terms of A016064 (where c = m+2) except for the first term, 1 (case with zero area).
Note that in all cases c is odd and m+2 <= c < 2m+1.

			First triangle has sides (3,4,5) and area 6.
2nd triangle has sides (9,10,17) and area 36.
3rd triangle has sides (13,14,15) and area 84.

Cf. A083875, A016064, A224301, A227003, A239978, A227166.

re=Reap[Do[a=m;b=m+1;Do[s=(a+b+c)/2;area=Sqrt[s(s-a)(s-b)(s-c)];If[IntegerQ[area],Sow[{a,b,c,area}];Break[]],{c,2m-1,m+2,-2 }],{m,3,2000}]][[2,1]];#[[1]]&/@ re

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A227166 Areas of indecomposable non-Pythagorean primitive integer Heronian triangles, sorted increasingly.

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