A070091 -id:A070091 - cp's OEIS frontend

A070084 Greatest common divisor of sides of integer triangles [A070080(n), A070081(n), A070082(n)], sorted by perimeter, sides lexicographically ordered.

1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 2, 1, 2, 1, 1, 1

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n)>1 iff there exists a smaller similar triangle [A070080(k), A070081(k), A070082(k)] with kA070080(n)=A070080(k)*a(n), A070081(n)=A070081(k)*a(n) and A070082(n)=A070082(k)*a(n).

R. Zumkeller, Integer-sided triangles

Cf. A051493, A005044, A070091, A070094, A070102, A070109, A070110, A070113, A070116, A070119, A070128, A070137.

maxPer = 22; maxSide = Floor[(maxPer - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPer^3 + a*maxPer^2 + b*maxPer + c; triangles = Reap[Do[If[a + b + c <= maxPer && 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]]; GCD @@@ Sort[triangles, order[#1] < order[#2] &] (* Jean-François Alcover, May 27 2013 *)

a(n) = GCD(A070080(n), A070081(n), A070082(n)).

A051493 Triangles with perimeter n and relatively prime integer side lengths.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 4, 2, 5, 2, 5, 4, 8, 4, 10, 6, 9, 6, 14, 8, 15, 9, 16, 12, 21, 11, 24, 16, 22, 16, 27, 18, 33, 20, 31, 24, 40, 23, 44, 30, 39, 30, 52, 32, 54, 35, 52, 42, 65, 38, 65, 48, 64, 49, 80, 48, 85, 56, 77, 64, 90, 58, 102, 72, 93, 69, 114, 72, 120, 81

Offset: 1

Neil Fernandez

nonn

From Peter Munn, Jul 26 2017: (Start)
The triangles that meet the conditions are listed by nondecreasing n in A070110.
Without the requirement for relatively prime side lengths, this sequence becomes A005044.
Counting the triangles by longest side instead of perimeter, this sequence becomes A123323.
a(n) = A070094(n) + A070102(n) + A070109(n).
(End)

			There are 3 triangles with integer-length sides and perimeter 9: 1-4-4, 2-3-4, 3-3-3. 3-3-3 is omitted because isomorphic to 1-1-1, so a(9)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A005044, A057887, A070110, A123323.

Equivalent sequences, restricted to subsets: A070091 (isosceles), A070094 (acute), A070102 (obtuse), A070109 (right-angled), A070138 (with integer area), A070202 (with integer inradius).

nmax = 100;
A005044[n_] := Quotient[n^2 + 6n Mod[n, 2] + 24, 48];
A = Array[A005044, nmax];
mob[m_, n_] := If[ Mod[m, n] == 0, MoebiusMu[m/n], 0];
Reap[Do[Sow[Sum[mob[n, d] A[[d]], {d, 1, n}]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Oct 05 2021 *)

Moebius transform of A005044.

Corrected and extended with formula by Christian G. Bower, Nov 15 1999

Formula updated due to change to referenced sequence, and definition clarified by Peter Munn, Jul 26 2017

A070099 Number of integer triangles with perimeter n and relatively prime side lengths which are acute and isosceles.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 4, 2, 2, 2, 5, 1, 4, 2, 4, 3, 6, 2, 6, 3, 4, 3, 5, 3, 8, 3, 4, 3, 8, 3, 9, 5, 5, 4, 10, 3, 9, 4, 6, 5, 11, 4, 8, 5, 7, 6, 12, 3, 13, 6, 8, 7, 9, 4, 14, 7, 8, 5, 15, 5, 15, 7, 9, 8, 13, 6, 16, 6, 11, 8, 17, 5, 13

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A059169, A051493, A070091, A070094, A070098, A070084, A070125.

A070107 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 1, 3, 2, 1, 1, 1, 0, 4, 2, 2, 2, 4, 1, 3, 2, 3, 2, 4, 1

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A059169, A051493, A070091, A070102, A070106, A070084, A070134.

A070116 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 19, 22, 23, 27, 28, 32, 35, 39, 40, 43, 46, 47, 51, 52, 55, 58, 61, 63, 64, 65, 72, 73, 81, 88, 94, 95, 98, 103, 104, 107, 108, 109, 118, 121, 124, 133, 135, 136, 140, 146, 150, 151, 159, 163, 166

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(10)=15: [A070080(15), A070081(15), A070082(15)]=[3<4=4], A070084(15)=gcd(3,4,4)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070091, A070125, A070134, A070115, A070110.

cp's OEIS Frontend

A070084 Greatest common divisor of sides of integer triangles [A070080(n), A070081(n), A070082(n)], sorted by perimeter, sides lexicographically ordered.

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A051493 Triangles with perimeter n and relatively prime integer side lengths.

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A070099 Number of integer triangles with perimeter n and relatively prime side lengths which are acute and isosceles.

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A070107 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and isosceles.

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A070116 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with relatively prime side lengths.

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