A020884 -id:A020884 - cp's OEIS frontend

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A020885 Ordered areas (divided by 6) of primitive Pythagorean triangles (with multiple entries).

1, 5, 10, 14, 30, 35, 35, 55, 84, 91, 105, 140, 154, 165, 204, 220, 231, 260, 285, 286, 385, 390, 429, 455, 455, 506, 595, 650, 680, 715, 770, 819, 836, 935, 969, 1015, 1105, 1190, 1240, 1309, 1326, 1330, 1330, 1495, 1496, 1615, 1729, 1771, 1785, 1820, 1925

Offset: 1

Clark Kimberling

nonn

Since squares are 0 or 1 under both mod 3 and mod 4, for the Pythagorean equation A^2 + B^2 = C^2 to hold, each of 3 and 4 divides either of leg A or leg B, so that area A*B/2 is divisible by 3*4/2 = 6. - Lekraj Beedassy, Apr 30 2004
From Wolfdieter Lang, Jun 14 2015: (Start)
This sequence gives the area/6 (in some squared length unit) of primitive Pythagorean triangles with multiplicities modulo leg exchange. See the example.
This sequence also gives Fibonacci's congruous numbers divided by 24, with multiplicities and ordered nondecreasingly. See A258150.
(End)
It appears that this sequence gives the list of dimensions of irreducible unitary representations of the Lie group SO(5). - Antoine Bourget, Mar 30 2022

			a(6) = a(7) = 35 from the two Pythagorean triangles (A,B,C) = (21, 20, 29)  and (35, 12, 37) with area 210. Triangles (20, 21, 29) and (12, 35, 37) are not counted (leg exchange). - _Wolfdieter Lang_, Jun 14 2015

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020882, A020883, A020884, A020886.

Take[Sort[(Times@@#)/12&/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@ Select[ Subsets[Range[1,41,2],{2}],GCD@@#==1&])],60] (* Harvey P. Dale, Feb 27 2012 *)

a(n) = A024406(n)/6.

Extended and corrected by David W. Wilson

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

A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 1

Len Smiley, Nov 01 2005

This sequence might also be called "The Non-Pythagorean integers" since no primitive Pythagorean triangle (PPT) exists containing them. Numbers of the form 4n-2 cannot be a leg or hypotenuse of PPT [a,b,c]. This excludes all even members of the present sequence. Integers 1 and zero are excluded because they form a 'degenerate triangle' with angles = 0. Compare A125667. - H. Lee Price, Feb 02 2007

Besides the first term this sequence is the denominator of Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - Mohammad K. Azarian, Oct 14 2011

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A125667. Complement of the union of {1}, A020882, A020883 and A020884.

Cf. A016825, A161718.

Table[If[n == 1, 1, 4n - 6], {n, 60}] (* Robert G. Wilson v, Nov 04 2005 *)

a(n) = 4*n-6, n>=2.
a(n) = A016825(n-2), n>1. - R. J. Mathar, Aug 18 2008
G.f.: x(1+3x^2)/(1-x)^2. - R. J. Mathar, Nov 10 2008
a(n^2 - 2n + 3)/2 = Sum_{i=1..n} a(i). - Ivan N. Ianakiev, Apr 24 2013
a(n) = 2*a(n-1) - a(n-2), n>3. - Rick L. Shepherd, Jul 06 2017
a(n) = |A161718(n-1)| = (-1)^(n-1)*A161718(n-1), n>0. - Rick L. Shepherd, Jul 06 2017
E.g.f.: 3*(x + 2) + exp(x)*(4*x - 6). - Stefano Spezia, Feb 02 2023

A024359 Number of primitive Pythagorean triangles with short leg n.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times A takes value n.
Number of times n occurs in A020884.
a(A139544(n)) = 0; a(A024352(n)) > 0. - Reinhard Zumkeller, Nov 09 2012

T. D. Noe, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020884, A024352, A024360, A024361, A132404 (where records occur), A139544.

a024359_list = f 0 1 a020884_list where
   f c u vs'@(v:vs) | u == v = f (c + 1) u vs
                    | u /= v = c : f 0 (u + 1) vs'
-- Reinhard Zumkeller, Nov 09 2012

solns[a_] := Module[{b, c, soln}, soln = Reduce[a^2 + b^2 == c^2 && a < b && c > 0 && GCD[a, b, c] == 1, {b, c}, Integers]; If[soln === False, 0, If[soln[[1, 1]] === b, 1, Length[soln]]]]; Table[solns[n], {n, 100}]
(* Second program: *)
a[n_] := Module[{s = 0, b, c, d, g}, Do[g = Quotient[n^2, d]; If[d <= g && Mod[d+g, 2] == 0, c = Quotient[d+g, 2]; b = g-c; If[n < b && GCD[b, c] == 1, s++]], {d, Divisors[n^2]}]; s]; Array[a, 100] (* Jean-François Alcover, Apr 27 2019, from PARI *)

nppt(a) = {
  my(s=0, b, c, d, g);
  fordiv(a^2, d,
    g=a^2\d;
    if(d<=g && (d+g)%2==0,
      c=(d+g)\2; b=g-c;
      if(aColin Barker, Oct 25 2015

Formula

a(n) = A024361(n) - A024360(n). - Ray Chandler, Feb 03 2020

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.

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

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Author

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

Keywords

nonn

Comments

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

Crossrefs

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

Programs

Mathematica

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

A063011 Ordered products of the sides of primitive Pythagorean triangles.

Original entry on oeis.org

60, 780, 2040, 4200, 12180, 14760, 15540, 40260, 65520, 66780, 92820, 120120, 189840, 192720, 199980, 235620, 277680, 354960, 453960, 497640, 595140, 619020, 643500, 1021020, 1063860, 1075620, 1265880, 1484340, 1609080, 1761540

Offset: 1

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Author

Henry Bottomley, Jul 26 2001

Keywords

nonn

Comments

It is an open question whether any two distinct Pythagorean triples can have the same product of their sides.

Examples

a(1)=3*4*5=60; a(2)=5*12*13=780 (rather than 6*8*10=480, which would not be primitive).

Crossrefs

Cf. A020882, A020883, A020884, A020885, A020886, A057096.

Programs

Mathematica

k=17000000;lst={};Do[Do[If[IntegerQ[a=Sqrt[c^2-b^2]]&&GCD[a,b,c]==1,If[a>=b,Break[]];x=a*b*c;If[x<=k,AppendTo[lst,x]]],{b,c-1,4,-1}],{c,5,700,1}];Union@lst (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009 *) With[{nn=50},Take[(Times@@#)Sqrt[#[[1]]^2+#[[2]]^2]&/@Union[Sort/@ ({Times@@#, (Last[#]^2-First[#]^2)/2}&/@(Select[Subsets[Range[1,nn+1,2],{2}],GCD@@#==1&]))]//Union,nn]] (* Harvey P. Dale, Jun 08 2018 *)

A081874 Ordered odd short legs of primitive Pythagorean triangles.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 33, 35, 37, 39, 39, 41, 43, 45, 47, 49, 51, 51, 53, 55, 57, 57, 59, 61, 63, 65, 65, 67, 69, 69, 71, 73, 75, 75, 77, 79, 81, 83, 85, 85, 87, 87, 89, 91, 93, 93, 95, 95, 97, 99, 101, 103, 105, 105, 105, 107, 109, 111

Offset: 1

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Author

Lekraj Beedassy, Apr 23 2003

Keywords

nonn

Comments

Sequence comprises all odd values greater than 1. In particular, primitive Pythagorean triangles with consecutive longer sides have long and short legs 2n(n+1) {A046092(n)} and 2n+1 {A005408(n)} respectively.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A020884, A046092, A005408.

Extensions

Corrected and extended by Ray Chandler, Oct 29 2003

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=2*p*q, 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

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Author

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

Keywords

nonn

Comments

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

Crossrefs

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

Programs

Mathematica

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

A235598 Begin with a(0) = 3. Let a(n) for n > 0 be the smallest positive integer not yet in the sequence which forms part of a Pythagorean triple when paired with a(n-1).

Original entry on oeis.org

3, 4, 5, 12, 9, 15, 8, 6, 10, 24, 7, 25, 20, 16, 30, 18, 80, 39, 36, 27, 45, 28, 21, 29, 420, 65, 33, 44, 55, 48, 14, 50, 40, 32, 60, 11, 61, 1860, 341, 541, 146340, 15447, 20596, 25745, 32208, 2540, 1524, 635, 381, 508, 16125, 4515, 936, 75, 72, 54, 90, 56

Offset: 0

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Author

Jack Brennen, Dec 26 2013

Keywords

nonn
nice

Comments

Is the sequence infinite? Can it "paint itself into a corner" at any point? Note that picking any starting point >= 5 seems to lead to a finite sequence ending in 5,3,4. For example, starting with 6 we get 6,8,10,24,7,25,15,9,12,5,3,4, stop (A235599).

By beginning with 3 or 4, we make sure that the 5,3,4 dead-end is never available.

If infinite, is it a permutation of the integers >= 3? This seems likely. Proving it doesn't seem easy though.

Comment from Jim Nastos, Dec 30 2013: Your question about whether the sequence can 'paint itself into a corner' is essentially asking if the Pythagorean graph has a Hamiltonian path. As far as I know, the questions in the Cooper-Poirel paper (see link) are still unanswered. They ask whether the graph is k-colorable with a finite k, or whether it is even connected (sort of equivalent to your question of whether it is a permutation of the integers >=3).

Lars Blomberg has computed the sequence out to 3 million terms without finding a dead end.

Position of k>2: 0, 1, 2, 7, 10, 6, 4, 8, 35, 3, 67, 30, 5, 13, 89, 15, 143, 12, 22, 118, 385, 9, 11, ..., see A236243. - Robert G. Wilson v, Jan 17 2014

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..10000

J. Cooper and C. Poirel, Note on the Pythagorean Triple System

Crossrefs

Cf. A020882, A020883, A020884, A235599, A236243, A236244.

Programs

Mathematica

f[s_List] := Block[{n = s[[-1]]}, sol = Solve[ x^2 + y^2 == z^2 && x > 0 && y > 0 && z > 0 && (x == n || z == n), {x, y, z}, Integers]; Append[s, Min[ Complement[ Union[ Extract[sol, Position[ sol, Integer]]], s]]]]; lst = Nest[f, {3}, 250] (* _Robert G. Wilson v, Jan 17 2014 *)

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A020885 Ordered areas (divided by 6) of primitive Pythagorean triangles (with multiple entries).

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Mathematica

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.

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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.

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Crossrefs

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Mathematica

Extensions

A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.

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Programs

Mathematica

Formula

A024359 Number of primitive Pythagorean triangles with short leg n.

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Haskell

Mathematica

PARI

Formula

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.

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Mathematica

A063011 Ordered products of the sides of primitive Pythagorean triangles.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A081874 Ordered odd short legs of primitive Pythagorean triangles.

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Author

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Comments

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Crossrefs

Extensions

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=2*p*q, s=a+b+c, s-+1 are primes.

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.