A068964 -id:A068964 - cp's OEIS frontend

A068966 Areas of integer Heronian triangles [prime(A068964(n)), prime(A068964(n)+1), A068965(n)].

6, 66, 3204, 6810, 37716, 72006, 182430, 532236, 370614, 2155134, 3203694, 6353634, 51712890, 42020844, 28698786, 33163770, 55637466, 125033580, 172985436, 105470250, 151375626, 178631034, 185921166, 217064574, 939603126, 376267326, 1812742734, 2193232470, 853918566

Offset: 1

Reinhard Zumkeller, Mar 30 2002

nonn

			A068964(3) = 24: [prime(24), prime(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 a(3) = 3204.

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

Cf. A046131, A055595, A068969.

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

A068967 Numbers that define integer Heronian triangles [a(n), prime(a(n)), A068968(n)] with area A068969(n).

Original entry on oeis.org

3, 12, 25, 30, 37, 51, 60, 65, 65, 100, 116, 260, 273, 761, 761, 825, 884, 1092, 1476, 1695, 1695, 1752, 2415, 3048, 3552, 4045, 4134, 4264, 5429, 6357, 7267, 9316, 14027, 16095, 18850, 21576, 29925, 30261, 38164, 40562, 45982, 45982, 51805, 51805, 82000, 82000

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 A068969(5) = 1800.

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

Cf. A068968, A068969, A068964.

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 &]]; (Join @@ Parallelize@ Array[zz, 4300])[[All, 1]] (* Giovanni Resta, Apr 20 2020 *)

Erroneous term 2374 removed and more terms from Giovanni Resta, Apr 20 2020

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A068966 Areas of integer Heronian triangles [prime(A068964(n)), prime(A068964(n)+1), A068965(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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A068967 Numbers that define integer Heronian triangles [a(n), prime(a(n)), A068968(n)] with area A068969(n).

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