A227166 -id:A227166 - cp's OEIS frontend

A239978 Areas of indecomposable primitive integer Heronian triangles (including primitive Pythagorean triangles), in increasing order.

6, 30, 60, 72, 84, 126, 168, 180, 210, 210, 252, 252, 288, 330, 336, 336, 396, 396, 420, 420, 420, 420, 456, 462, 504, 528, 528, 546, 624, 630, 714, 720, 720, 756, 792, 798, 840, 840, 840, 840, 840, 864, 924, 924, 924, 924, 924, 936, 990, 990, 1008

Offset: 1

Frank M Jackson, Mar 30 2014

nonn

An indecomposable Heronian triangle is a Heronian triangle that cannot be split into two Pythagorean triangles. In other words, it has no integer altitude that is not a side of the triangle. Note that all primitive Pythagorean triangles are indecomposable.

See comments in A227003 about the Mathematica program below to ensure that all primitive Heronian areas up to 1008 are captured.

			a(5) = 84 as this is the fifth ordered area of an indecomposable primitive Heronian triangle. The triple is (7,24,25) and it is Pythagorean.

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

Cf. A083875, A224301, A227003, A227166.

nn=1008; 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])||(c^2+b^2==a^2)), AppendTo[lst, Sqrt[area2]]]], {a,3,nn}, {b,a}, {c,b}]; Sort@Select[lst, #<=nn &] (*using T. D. Noe's program A083875*)

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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A239978 Areas of indecomposable primitive integer Heronian triangles (including primitive Pythagorean triangles), in increasing order.

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