A051493 -id:A051493 - 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)).

A070138 Number of integer triangles with an integer area having relatively prime sides a, b and c such that a + b + c = n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 3

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

From Peter Kagey, Jan 30 2018: (Start)
a(k) > 0 if and only if k is in A096468.
Records appear at indices 12, 36, 54, 84, 98, 162, 242, 338, 484, 578, ....
a(2k - 1) = 0 for all integers k > 0.
(End)

Peter Kagey, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A051493, A051516, A070088, A070109, A070143, A096468, A239246.

Corrected by T. D. Noe, Jun 17 2004

A070110 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 27, 28, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 72, 73, 74, 75, 77

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

A070084(a(k)) = gcd(A070080(a(k)), A070081(a(k)), A070082(a(k))) = 1;

all integer triangles [A070080(a(k)), A070081(a(k)), A070082(a(k))] are mutually nonisomorphic.

			13 is a term: [A070080(13), A070081(13), A070082(13)]=[2,4,5], A070084(13)=gcd(2,4,5)=1.

Jean-François Alcover, Table of n, a(n) for n = 1..789
Reinhard Zumkeller, Integer-sided triangles

Cf. A051493, A070113, A070116, A070119, A070122, A070125, A070128, A070131, A070134, A070137, A070084.

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_} /; GCD[a, b, c] == 1] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070109 Number of right integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Right integer triangles have integer areas: see A070142, A051516.

a(n) is nonzero iff n is in A024364.

			For n=30 there are A005044(30) = 19 integer triangles; only one is right: 5+12+13 = 30, 5^2+12^2 = 13^2; therefore a(30) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (obtained from the b-file of A078926)
Eric Weisstein's World of Mathematics, Right Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triples.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A051493, A070093, A070101, A070138, A070084, A070137.

unitaryDivisors[n_] := Cases[Divisors[n], d_ /; GCD[d, n/d] == 1];
A078926[n_] := Count[unitaryDivisors[n], d_ /; OddQ[d] && Sqrt[n] < d < Sqrt[2n]];
a[n_] := If[EvenQ[n], A078926[n/2], 0];
Table[a[n], {n, 1, 1716}] (* Jean-François Alcover, Oct 04 2021 *)

a(n) = A078926(n/2) if n is even; a(n)=0 if n is odd.
a(n) = A051493(n) - A070094(n) - A070102(n).
a(n) <= A024155(n).

Secondary offset added by Antti Karttunen, Oct 07 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)).

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

Original entry on oeis.org

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.

A070102 Number of obtuse integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 1, 3, 2, 3, 2, 5, 3, 6, 2, 8, 5, 9, 5, 9, 6, 11, 6, 14, 9, 14, 9, 17, 11, 19, 12, 19, 15, 23, 13, 27, 18, 26, 16, 32, 20, 33, 21, 34, 26, 40, 23, 42, 29, 42, 29, 50, 32, 53, 35, 48, 41, 58, 37, 64, 45, 60, 42, 71

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051493(n) - A070094(n) - A070109(n).

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; only one of them is obtuse: 2^2+3^2<16=4^2 and GCD(2,3,4)=1, therefore a(9)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A051493, A070104, A070107, A070084, A070128.

A070091 Number of isosceles integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 1, 1, 3, 1, 3, 1, 2, 2, 4, 2, 5, 2, 2, 2, 6, 2, 5, 3, 5, 3, 7, 2, 8, 4, 4, 4, 6, 3, 9, 4, 6, 4, 10, 4, 11, 5, 6, 5, 12, 4, 10, 5, 8, 6, 13, 4, 10, 6, 8, 7, 15, 4, 15, 7, 10, 8, 12, 6, 17, 8, 10, 6, 18, 6, 18, 9, 10, 9, 14, 6, 20, 8, 13

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051493(n) - A005044(n-6).

			For n=15 there are A005044(15)=7 integer triangles: [1,7,7], [2,6,7], [3,5,7], [3,6,6], [4,4,7], [4,5,6] and [5,5,5]: four are isosceles: [1<7=7], [3<6=6], [4=4<7] and [5=5=5], but GCD(3,6,6)>1 and GCD(5,5,5)>1, therefore a(15)=2.

Reinhard Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A059169, A070099, A070107, A070084, A070116.

m = 81 (* 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]] &] ;
a[n_] := Count[triangles, t_ /; Total[t] == n && Length[Union[t]] < 3 && GCD @@ t == 1];
Table[a[n], {n, 1, m}] (* Jean-François Alcover, Oct 05 2021 *)

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.

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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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A070138 Number of integer triangles with an integer area having relatively prime sides a, b and c such that a + b + c = n.

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A070110 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with relatively prime side lengths.

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A070109 Number of right integer triangles with perimeter n and relatively prime 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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A070094 Number of acute integer triangles with perimeter n and relatively prime side lengths.

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A070102 Number of obtuse integer triangles with perimeter n and relatively prime side lengths.

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