A024364 -id:A024364 - cp's OEIS frontend

A010814 Perimeters of integer-sided right triangles.

12, 24, 30, 36, 40, 48, 56, 60, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 168, 176, 180, 182, 192, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 276, 280, 286, 288, 300, 306, 308, 312, 320, 324

Offset: 1

Ben Manvel (manvel(AT)lagrange.math.colostate.edu)

A number k is in this sequence iff k is a multiple of some term in A024364.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Ron Knott, Pythagorean Triples and Online Calculators.
Index entries related to Pythagorean Triples.

Twice A005279.

Cf. A024364.

isA010814 := proc(an) local a::integer,b::integer,c::integer ; for c from 1 to floor(an/2) do for a from floor(c/sqrt(2)) to c-1 do if issqr(c^2-a^2) then b := sqrt(c^2-a^2) ; if a+b+c = an then RETURN(true) ; fi ; fi ; od ; od : RETURN(false) ; end : for n from 3 to 400 do if isA010814(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

lst={};Do[Do[If[IntegerQ[c=Sqrt[a^2+b^2]],AppendTo[lst,a+b+c]],{b,a-1,Floor[Sqrt[a]],-1}],{a,4,4*5!}];Take[Union@lst,100] (* Vladimir Joseph Stephan Orlovsky, Nov 23 2010 *)
q[n_] := EvenQ[n] && Module[{d = Divisors[n/2]}, AnyTrue[Range[3, Length[d]], d[[#]] < 2 * d[[#-1]] &]]; Select[Range[350], q] (* Amiram Eldar, Oct 19 2024 *)

select( {is_A010814(n)=n%2==0&&is_A005279(n\2)}, [1..333]) \\ M. F. Hasler, Mar 20 2025

More terms from Ray Chandler, Mar 13 2004

A020886 Ordered semiperimeters of primitive Pythagorean triangles.

Original entry on oeis.org

6, 15, 20, 28, 35, 42, 45, 63, 66, 72, 77, 88, 91, 99, 104, 110, 117, 120, 130, 143, 153, 156, 165, 170, 187, 190, 195, 204, 209, 210, 221, 228, 231, 238, 247, 255, 266, 272, 273, 276, 285, 299, 304, 322, 323, 325, 336, 342, 345, 350, 357, 368, 378, 391, 399

Offset: 1

Clark Kimberling

nonn

k is in this sequence iff A078926(k) > 0.
Also, ordered sides c of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c (A343893). - Bernard Schott, May 06 2021
a(n) are the ordered radii of inscribed circles in squares, from which the tangents to the circles are cut off by primitive Pythagorean triangles. - Alexander M. Domashenko, Oct 17 2024

Ray Chandler, Table of n, a(n) for n = 1..10000 (duplicates removed by Sean A. Irvine)

Subsequence of A005279.
Cf. A024364, A078926, A020882, A020884, A020885.
Triangles with 2/a = 1/b + 1/c:  A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

isA020886 := proc(an) local r::integer,s::integer ; for r from floor((an/2)^(1/2)) to floor(an^(1/2)) do for s from r-1 to 1 by -2 do if r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 400 do if isA020886(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[ n/2^IntegerExponent[n, 2]]}];
Select[Range[1000], A078926[#]>0&] (* Jean-François Alcover, Mar 23 2020 *)

is(n,f=factor(n))=my(P=apply(i->f[i,1]^f[i,2],[2-n%2..#f~]),nn=2*n); forvec(v=vector(#P,i,[0,1]), my(d=prod(i=1,#v,P[i]^v[i]),d2=d^2); if(d2n, return(1))); 0
list(lim)=my(v=List()); forfactored(n=6,lim\1, if(is(n[1],n[2]), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Feb 03 2023

a(n) = A024364(n)/2.

A103606 Primitive Pythagorean triples in nondecreasing order of perimeter, with each triple in increasing order, and if perimeters coincide then increasing order of the even members.

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 8, 15, 17, 7, 24, 25, 20, 21, 29, 12, 35, 37, 9, 40, 41, 28, 45, 53, 11, 60, 61, 16, 63, 65, 33, 56, 65, 48, 55, 73, 13, 84, 85, 36, 77, 85, 39, 80, 89, 20, 99, 101, 65, 72, 97

Offset: 1

Alexandre Wajnberg, Mar 24 2005

The NAME was corrected by a proposal of Wolfdieter Lang. - Ralf Steiner, Sep 29 2019
The corresponding perimeters are given in A024364. - Wolfdieter Lang, Oct 06 2014
Note that the multiplicity of primitive Pythagorean triples (increasingly ordered) with perimeter P is not always 1. See A024408 for P numbers with multiplicity k >= 2, and the first example with k = 2 for P = 1716. - Wolfdieter Lang, Sep 24 2019

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, page 328.

Jean-François Alcover, Table of n, a(n) for n = 1..3975
Ron Knott, The Pythagorean triples.
Michael Penn, Number Theory | Primitive Pythagorean Triples, Youtube video, 2019.

Subsequence of A103605.

Cf. A024364, A024408.

A103605 = Cases[Import["https://oeis.org/A103605/b103605.txt", "Table"], {, }][[All, 2]];
SortBy[Select[Partition[A103605, 3], GCD @@ # == 1&], {#[[1]] + #[[2]] + #[[3]]&, If[EvenQ[#[[1]]], #[[1]], #[[2]]]&}] // Flatten (* Jean-François Alcover, May 26 2020 *)

Corrected at the suggestion of Ralf Steiner by Wolfdieter Lang, Sep 24 2019

Errors in b-file noticed by Kevin Ryde corrected by Jean-François Alcover, May 26 2020

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

A009096 Ordered perimeters of Pythagorean triangles, listed with multiplicity.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 56, 60, 60, 70, 72, 80, 84, 84, 90, 90, 96, 108, 112, 120, 120, 120, 126, 132, 132, 140, 144, 144, 150, 154, 156, 160, 168, 168, 168, 176, 180, 180, 180, 182, 192, 198, 200, 204, 208, 210, 210, 216, 220, 224, 228, 234, 240, 240, 240, 240, 252, 252

Offset: 1

David W. Wilson

nonn

			The perimeters are listed with multiplicity, for example a(8) = a(9) = 60 = 15 + 20 + 25 = 10 + 24 + 26. - _M. F. Hasler_, Jul 04 2025

Cf. A103605 (the corresponding triples), A024364 (perimeters of primitive Pythagorean triangles).

a(n) = A103605(3n) + A103605(3n-1) + A103605(3n-2). - M. F. Hasler, Jul 04 2025

Name clarified by M. F. Hasler, Jul 04 2025

A155171 Numbers p such that if q = p+1 then (a = q^2-p^2, b = 2pq, c = q^2 + p^2) is a primitive Pythagorean triple with s-1 and s+1 primes, where s = a+b+c.

Original entry on oeis.org

1, 2, 7, 10, 20, 29, 44, 50, 65, 70, 76, 77, 101, 104, 107, 115, 154, 175, 197, 202, 226, 227, 247, 275, 371, 380, 412, 457, 490, 500, 574, 596, 647, 671, 682, 710, 764, 829, 926, 1052, 1085, 1102, 1127, 1186, 1204, 1205, 1225, 1256, 1280, 1324, 1325, 1331

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

			p=1,q=2,a=3,b=4,c=5,s=12-+1 primes.

Cf. A020882, A020886, A020884, A020883, A024364, A024406

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,n]],{n,8!}];lst

Definition edited by N. J. A. Sloane, Jul 19 2022

A155173 Short leg A of primitive Pythagorean triangles such that perimeter s is average of twin prime pairs, q=p+1, A=q^2-p^2, C=q^2+p^2, B=2pq, s=A+B+C; s -/+ 1 are primes.

Original entry on oeis.org

3, 5, 15, 21, 41, 59, 89, 101, 131, 141, 153, 155, 203, 209, 215, 231, 309, 351, 395, 405, 453, 455, 495, 551, 743, 761, 825, 915, 981, 1001, 1149, 1193, 1295, 1343, 1365, 1421, 1529, 1659, 1853, 2105, 2171, 2205, 2255, 2373, 2409, 2411, 2451, 2513, 2561, 2649

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

With p=1, then q=2,a=3,b=4,c=5, and s=12-+1 (11, 13) both primes.

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171.

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,a]],{n,8!}];lst

Name edited by Zak Seidov, Mar 21 2014

A155174 Long leg B of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

Original entry on oeis.org

4, 12, 112, 220, 840, 1740, 3960, 5100, 8580, 9940, 11704, 12012, 20604, 21840, 23112, 26680, 47740, 61600, 78012, 82012, 102604, 103512, 122512, 151800, 276024, 289560, 340312, 418612, 481180, 501000, 660100, 711624, 838512, 901824, 931612

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,b]],{n,8!}];lst

A155175 Hypotenuse C of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

Original entry on oeis.org

5, 13, 113, 221, 841, 1741, 3961, 5101, 8581, 9941, 11705, 12013, 20605, 21841, 23113, 26681, 47741, 61601, 78013, 82013, 102605, 103513, 122513, 151801, 276025, 289561, 340313, 418613, 481181, 501001, 660101, 711625, 838513, 901825, 931613

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,c]],{n,8!}];lst

A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 5, 40, 77, 287, 590, 1335, 1717, 2882, 3337, 3927, 4030, 6902, 7315, 7740, 8932, 15965, 20592, 26070, 27405, 34277, 34580, 40920, 50692, 92132, 96647, 113575, 139690, 160557, 167167, 220225, 237407, 279720, 300832, 310765, 336777, 389895

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,s/6]],{n,8!}];lst

cp's OEIS Frontend

A010814 Perimeters of integer-sided right triangles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A020886 Ordered semiperimeters of primitive Pythagorean triangles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A103606 Primitive Pythagorean triples in nondecreasing order of perimeter, with each triple in increasing order, and if perimeters coincide then increasing order of the even members.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A009096 Ordered perimeters of Pythagorean triangles, listed with multiplicity.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A155171 Numbers p such that if q = p+1 then (a = q^2-p^2, b = 2*p*q, c = q^2 + p^2) is a primitive Pythagorean triple with s-1 and s+1 primes, where s = a+b+c.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A155173 Short leg A of primitive Pythagorean triangles such that perimeter s is average of twin prime pairs, q=p+1, A=q^2-p^2, C=q^2+p^2, B=2*p*q, s=A+B+C; s -/+ 1 are primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A155174 Long leg B of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

Views

Author

Keywords

Comments

A155171 Numbers p such that if q = p+1 then (a = q^2-p^2, b = 2pq, c = q^2 + p^2) is a primitive Pythagorean triple with s-1 and s+1 primes, where s = a+b+c.

A155173 Short leg A of primitive Pythagorean triangles such that perimeter s is average of twin prime pairs, q=p+1, A=q^2-p^2, C=q^2+p^2, B=2pq, s=A+B+C; s -/+ 1 are primes.

A155174 Long leg B of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

A155175 Hypotenuse C of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.