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A070148 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer Heronian triangle having triangular area. 2
17, 368, 659, 972, 1156, 1599, 1971, 2555, 2574, 3746, 3818, 4298, 4330, 5374, 14325, 14414, 15004, 15943, 16451, 19475, 19615, 24013, 24051, 33950, 63593, 71630, 75052, 79286, 79670, 79921, 84183, 90187, 93290 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Jean-François Alcover, Table of n, a(n) for n = 1..89

Eric Weisstein's World of Mathematics, Heronian Triangle.

Eric Weisstein's World of Mathematics, Right Triangle.

Reinhard Zumkeller, Integer-sided triangles

EXAMPLE

17 is a term: [A070080(17), A070081(17), A070082(17)] = [3,4,5]: A070086(52)=6.

MATHEMATICA

m = 500 (* max perimeter *);

sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];

triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];

area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];

Position[triangles, {a_, b_, c_} /; IntegerQ[area[{a, b, c}]] && IntegerQ[Sqrt[1 + 8 area[{a, b, c}]]]] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

CROSSREFS

Cf. A070142, A000217.

Sequence in context: A222678 A293691 A002197 * A097499 A132541 A159244

Adjacent sequences:  A070145 A070146 A070147 * A070149 A070150 A070151

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 05 2002

STATUS

approved

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Last modified May 17 13:48 EDT 2022. Contains 353746 sequences. (Running on oeis4.)