A070096 -id:A070096 - cp's OEIS frontend

A070094 Number of acute integer triangles with perimeter n and relatively prime side lengths.

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 5, 2, 5, 3, 3, 4, 6, 3, 6, 4, 7, 6, 10, 4, 10, 7, 8, 7, 10, 7, 14, 8, 12, 8, 17, 10, 17, 12, 13, 14, 20, 12, 21, 14, 18, 16, 25, 15, 23, 18, 22, 20, 30, 16, 32, 21, 29, 23, 32, 21, 38, 27, 33, 26, 43, 25

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051493(n) - A070102(n) - A070109(n).

			For n=10 there are A005044(10) = 2 integer triangles: [2,4,4] and [3,3,4]; both are acute, but GCD(2,4,4)>1, therefore a(9) = 1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A070096, A070099, A070084, A070119.

A070122 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

33, 45, 53, 60, 70, 83, 90, 92, 106, 114, 119, 132, 134, 142, 148, 162, 165, 168, 175, 181, 183, 197, 200, 203, 204, 218, 224, 237, 240, 245, 247, 261, 264, 267, 268, 282, 290, 293, 296, 309, 316, 317, 319, 333, 341, 345, 348

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			70 is a term because [A070080(70), A070081(70), A070082(70)]=[5<7<8], A070084(70)=gcd(5,7,8)=1, A070085(70)=5^2+7^2-8^2=25+49-64=10>0.

Jean-François Alcover, Table of n, a(n) for n = 1..229
Reinhard Zumkeller, Integer-sided triangles

Cf. A070096, A070118, A070110, A070112.

m = 55 (* 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]]&];
Position[triangles, {a_, b_, c_} /; a < b < c && GCD[a, b, c] == 1 && a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

cp's OEIS Frontend

A070094 Number of acute integer triangles with perimeter n and relatively prime side lengths.

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