A068966 -id:A068966 - cp's OEIS frontend

A068964 Numbers that define integer Heronian triangles [prime(a(n)), prime(a(n)+1), A068965(n)] with area A068966(n).

2, 5, 24, 49, 87, 182, 307, 379, 458, 677, 1618, 2427, 4397, 4944, 5969, 6496, 8871, 9526, 11580, 13063, 16243, 17970, 18400, 20225, 25271, 28275, 37730, 42387, 46597, 48681, 50329, 56586, 58859, 61930, 64289, 65078, 70776, 78360, 80133, 83638, 83640, 115956, 118712

Offset: 1

Reinhard Zumkeller, Mar 30 2002

nonn

			a(3) = 24: [A000040(24), A000040(25), A068965(3)] = [89,97,170], with s = (89+97+170)/2 = 178: Area^2 = s*(s-89)*(s-81)*(s-170) = 178*89*81*8 = 10265616 = 3204*3204, therefore A068966(3) = 3204.

Giovanni Resta, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Heronian Triangle.

Terms a(13) and beyond from Giovanni Resta, Apr 20 2020

A068965 Sides of integer Heronian triangles [prime(A068964(n)), prime(A068964(n)+1), a(n)] with area A068966(n).

Original entry on oeis.org

4, 20, 170, 452, 890, 2180, 4052, 5210, 6500, 10100, 27380, 43220, 84068, 95930, 118100, 130052, 183620, 198458, 246410, 281252, 357860, 399620, 410420, 455060, 581060, 656660, 900500, 1022468, 1134020, 1188890, 1232468, 1401140, 1462058, 1545290, 1609220, 1630820

Offset: 1

Reinhard Zumkeller, Mar 30 2002

nonn

			A068964(3) = 24: [A000040(24), A000040(25), a(3)] = [89,97,170], with s = (89+97+170)/2 = 178: Area^2 = s*(s-89)*(s-81)*(s-170) = 178*89*81*8 = 10265616 = 3204*3204, therefore A068966(3) = 3204.

Giovanni Resta, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Heronian Triangle.

Terms a(13) and beyond from Giovanni Resta, Apr 20 2020

A068969 Areas of integer Heronian triangles [A068967(n), prime(A068967(n)), A068968(n)].

Original entry on oeis.org

6, 210, 528, 1680, 1800, 5304, 6930, 3150, 4650, 21000, 32760, 69342, 53550, 2170560, 2200200, 2501070, 646800, 4777080, 4796550, 11865840, 12243840, 12863760, 18064200, 30510480, 3232320, 66023100, 70691400, 47977380, 144357720, 185560830, 156128700, 320843040

Offset: 1

Reinhard Zumkeller, Mar 30 2002

nonn

			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 a(5) = 1800.

Giovanni Resta, Table of n, a(n) for n = 1..120
Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Heron's Formula.

Cf. A046131, A055595, A068966.

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 &]]; area /@ Join @@ Parallelize@ Array[zz, 4300] (* Giovanni Resta, Apr 20 2020 *)

Erroneous term 25070627 removed and more terms from Giovanni Resta, Apr 20 2020

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A068964 Numbers that define integer Heronian triangles [prime(a(n)), prime(a(n)+1), A068965(n)] with area A068966(n).

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A068965 Sides of integer Heronian triangles [prime(A068964(n)), prime(A068964(n)+1), a(n)] with area A068966(n).

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A068969 Areas of integer Heronian triangles [A068967(n), prime(A068967(n)), A068968(n)].

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