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A070142 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area. 11
17, 39, 52, 116, 212, 252, 269, 368, 370, 372, 375, 493, 561, 587, 659, 839, 850, 862, 957, 972, 1156, 1186, 1196, 1204, 1297, 1582, 1599, 1629, 1912, 1920, 1955, 1971, 1988, 2115, 2352, 2555, 2574, 2713, 2774, 2778, 2790 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..41.

R. Zumkeller, Integer-sided triangles

Eric Weisstein's World of Mathematics, Heronian Triangle.

EXAMPLE

a(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

MATHEMATICA

maxPerim = 100; maxSide = Floor[(maxPerim - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPerim^3 + a*maxPerim^2 + b*maxPerim + c; triangles = Reap[ Do[ If[ a + b + c <= maxPerim && c - b < a < c + b && b - a < c < b + a && c - a < b < c + a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; stri = Sort[ triangles, order[#1] < order[#2]&]; area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p*(p - a)*(p - b)*(p - c)]]; Position[ stri, tri_ /; IntegerQ[area[tri]]] // Flatten (* Jean-Fran├žois Alcover, Feb 22 2013 *)

CROSSREFS

Cf. A070088, A070149, A070143, A070144, A070145, A051516, A069594, A069597, A070136, A070137, A070209.

Sequence in context: A259489 A226127 A156777 * A070143 A274555 A140679

Adjacent sequences:  A070139 A070140 A070141 * A070143 A070144 A070145

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 05 2002

STATUS

approved

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Last modified December 7 03:36 EST 2016. Contains 278838 sequences.