A070110 -id:A070110 - cp's OEIS frontend

A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

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

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

			Table begins (imprimitive triples are labeled i):
[1,1,1],
[2,2,1],
[2,2,2],i
[3,2,2],
[3,3,1],
[3,3,2],
[3,3,3],i
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,2],i
[4,4,3],
[4,4,4],i
[5,3,3],
...

Lars Blomberg, Table of n, a(n) for n = 1..9000

There are A002620(k+1) rows that begin with k.
The three columns are A316843, A316844, A316845.
A316849 is a compressed version.
See A316842 for primitive triples.
See A316851 and A316853 & A317182 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

for(i=1,6, for(j=1,i, for(k=1,j, if(j+k>i, print1(i,", ",j,", ",k,", "))))) \\ Hugo Pfoertner, Jan 25 2020

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

Original entry on oeis.org

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

A316842 Three-column table read by rows giving primitive integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i, gcd(i,j,k) = 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 1, 3, 3, 2, 4, 3, 2, 4, 3, 3, 4, 4, 1, 4, 4, 3, 5, 3, 3, 5, 4, 2, 5, 4, 3, 5, 4, 4, 5, 5, 1, 5, 5, 2, 5, 5, 3, 5, 5, 4, 6, 4, 3, 6, 5, 2, 6, 5, 3, 6, 5, 4, 6, 5, 5, 6, 6, 1, 6, 6, 5, 7, 4, 4, 7, 5, 3, 7, 5, 4, 7, 5, 5, 7, 6, 2, 7, 6, 3, 7, 6, 4, 7, 6, 5, 7, 6, 6, 7, 7, 1, 7, 7, 2, 7, 7, 3, 7, 7, 4, 7, 7, 5, 7, 7, 6, 8, 5, 4

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

			Table begins:
[1,1,1],
[2,2,1],
[3,2,2],
[3,3,1],
[3,3,2],
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,3],
[5,3,3],
[5,4,2],
...

Lars Blomberg, Table of n, a(n) for n = 1..9000

There are A123323(k) rows that begin with k.
The three columns are A316846, A316847, A316848.
A316850 is a compressed version.
See A316841 for all triples (including imprimitive triples).
See A316852 and A317181 & A317183 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

A123323 Number of integer-sided triangles with maximum side n, with sides relatively prime.

Original entry on oeis.org

1, 1, 3, 4, 8, 7, 15, 14, 21, 20, 35, 26, 48, 39, 52, 52, 80, 57, 99, 76, 102, 95, 143, 100, 160, 132, 171, 150, 224, 148, 255, 200, 250, 224, 300, 222, 360, 279, 348, 296, 440, 294, 483, 370, 444, 407, 575, 392, 609, 460, 592, 516, 728, 495, 740, 588, 738, 644

Offset: 1

Franklin T. Adams-Watters, Sep 25 2006

Number of triples a,b,c with a <= b <= c < a+b, gcd(a,b,c) = 1 and c = n.

Dropping the requirement for side lengths to be relatively prime this sequence becomes A002620 (with a different offset). See the Sep 2006 comment in A002620. - Peter Munn, Jul 26 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Stackexchange, Number of triples for which gcd(a,b,c)=1 and c=n, Feb 25 2014

Cf. A002620, A051493, A054875, A070110, A123324, A123325, A239246.

with(numtheory):
a:= n-> add(mobius(n/d)*floor((d+1)^2/4), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 23 2013

a[n_] := DivisorSum[n, Floor[(#+1)^2/4]*MoebiusMu[n/#]&]; Array[a, 60] (* Jean-François Alcover, Dec 07 2015 *)

A123323(n)=sumdiv(n,d,floor((d+1)^2/4)*moebius(n/d))

Moebius transform of b(n) = floor((n+1)^2/4).

G.f.: (G(x)+x-x^2)/2, where G(x) = Sum_{k >= 1} mobius(k)*x^k*(1+2*x^k-x^(2*k))/(1-x^k)^2/(1-x^(2*k)).

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 *)

A070125 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 6, 7, 11, 12, 15, 16, 19, 22, 23, 27, 28, 35, 39, 40, 43, 46, 47, 51, 55, 58, 63, 64, 65, 72, 73, 81, 88, 94, 95, 98, 103, 107, 108, 109, 121, 124, 135, 136, 140, 150, 151, 159, 166, 167, 170, 178, 185, 186, 189, 194, 201, 205

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(14)=22: [A070080(22), A070081(22), A070082(22)]=[3<5=5], A070084(22)=gcd(3,5,5)=1, A070085(22)=3^2+5^2-5^2=9>0.

R. Zumkeller, Integer-sided triangles

Cf. A070099, A070115, A070110, A070118.

A070131 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

8, 13, 20, 21, 25, 29, 30, 36, 37, 41, 42, 44, 49, 56, 57, 59, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 89, 96, 97, 99, 100, 101, 102, 105, 110, 111, 113, 115, 122, 123, 125, 126, 127, 128, 130, 131, 138, 141, 144, 147, 152, 153

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(20)=69: [A070080(69), A070081(69), A070082(69)]=[5<6<9], A070084(69)=gcd(5,6,9)=1, A070085(69)=5^2+6^2-9^2=25+36-81=-20<0.

R. Zumkeller, Integer-sided triangles

Cf. A070104, A070110, A070127, A070112.

A070134 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 14, 32, 52, 61, 104, 118, 133, 146, 163, 202, 242, 246, 266, 314, 342, 404, 437, 467, 472, 504, 542, 547, 577, 619, 625, 714, 757, 801, 807, 853, 907, 957, 1015, 1022, 1082, 1139, 1145, 1265, 1278, 1335, 1414, 1475

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(5)=61: [A070080(61), A070081(61), A070082(61)]=[5=5<9], A070084(69)=gcd(5,5,9)=1, A070085(61)=5^2+5^2-9^2=25+25-81=-31<0.

R. Zumkeller, Integer-sided triangles

Cf. A070107, A070110, A070127, A070115.

A070113 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 119, 122, 123, 125, 126

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			36 is a term [A070080(36), A070081(36), A070082(36)]=[3<6<7], A070084(36)=gcd(3,6,7)=1.

Jean-François Alcover, Table of n, a(n) for n = 1..596
Reinhard Zumkeller, Integer-sided triangles

Cf. A005044, A070122, A070131, A070112, A070110.

m = 50 (* 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] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

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A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

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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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A316842 Three-column table read by rows giving primitive integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i, gcd(i,j,k) = 1.

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A123323 Number of integer-sided triangles with maximum side n, with sides relatively prime.

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

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A070125 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle with relatively prime side lengths.

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A070131 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle with relatively prime side lengths.

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A070134 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with relatively prime side lengths.

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