A020882 -id:A020882 - cp's OEIS frontend

A024409 Hypotenuses of more than one primitive Pythagorean triangle.

65, 85, 145, 185, 205, 221, 265, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 629, 685, 689, 697, 725, 745, 785, 793, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1073, 1105, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313, 1325

Offset: 1

David W. Wilson

nonn

The subsequence allowing 4 different shapes is in A159781. [R. J. Mathar, Apr 12 2010]

A024362(a(n)) > 1. - Reinhard Zumkeller, Dec 02 2012

			65^2 = 16^2 + 63^2 = 33^2 + 56^2 (also = 25^2 + 60^2 = 39^2 + 52^2, but these are not primitive, with gcd = 5 resp. 13). Note that 65 = 1^2 + 8^2 = 4^2 + 7^2 is also the least integer > 1 to be a sum a^2 + b^2 with gcd(a,b) = 1 in two ways. - _M. F. Hasler_, May 18 2023

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020882, A120960, subsequence of A008846.

import Data.List (findIndices)
a024409 n = a024409_list !! (n-1)
a024409_list = map (+ 1) $ findIndices (> 1) a024362_list
-- Reinhard Zumkeller, Dec 02 2012

f[c_] := f[c] = Block[{a = 1, b, cnt = 0, lmt = Floor[ Sqrt[c^2/2]]}, While[b = Sqrt[c^2 - a^2]; a < lmt, If[IntegerQ@ b && GCD[a, b, c] == 1, cnt++]; a++]; cnt]Select[1 + 4 Range@ 335, f@# > 1 &] (* Robert G. Wilson v, Mar 16 2014 *)
Select[Tally[Sqrt[Total[#^2]]&/@Union[Sort/@({Times@@#,(Last[#]^2-First[ #]^2)/2}&/@(Select[Subsets[Range[1,71,2],{2}],GCD@@# == 1&]))]],#[[2]]> 1&][[All,1]]//Sort (* Harvey P. Dale, Sep 29 2018 *)

A014498 Varying radii of inscribed circles within primitive Pythagorean triples as a function of increasing values of hypotenuse.

Original entry on oeis.org

1, 2, 3, 3, 6, 5, 4, 10, 5, 12, 7, 15, 14, 6, 15, 20, 9, 21, 7, 18, 28, 11, 8, 21, 30, 35, 22, 9, 36, 24, 35, 13, 42, 33, 45, 10, 26, 40, 44, 15, 39, 11, 30, 45, 55, 56, 30, 63, 52, 12, 33, 66, 17, 63, 65, 72, 34, 13, 77, 60, 55, 70, 78, 19, 51, 14, 88, 39, 60, 77, 38, 91, 68, 90

Offset: 1

RALPH PETERSON (ralphp(AT)LIBRARY.NRL.NAVY.MIL)

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Saltire Software, Pythagorean Triples

For ordered values of (a+b-c)/2 see A020888.

Cf. A046086, A046087, A020882, A087459.

Arrange all primitive Pythagorean triples a, b, c by value of hypotenuse c, then by long leg b; for n-th value of c, sequence gives radius of largest inscribed circle, (a+b-c)/2.
a(n) = (A046086(n) + A046087(n) - A020882(n))/2 = A087459(n)/2.
a(n) = sqrt(A118961(n)*A118962(n)/2). - Lekraj Beedassy, May 07 2006

More terms from Asher Auel May 05 2000

Extended by Ray Chandler, Mar 09 2004

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

Original entry on oeis.org

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

A137409 Numbers that cannot be the value of 'C' in a primitive Pythagorean triple (A < B; A^2 + B^2 = C^2).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 14 2008

nonn

Complement of A008846. - R. J. Mathar, Aug 15 2010
A024362(a(n)) = 0. - Reinhard Zumkeller, Dec 02 2012
Except for the 1st term 1, complement of A004613. - Federico Provvedi, Jan 26 2019
After 1, numbers k for which A065338(k) > 1, i.e., after 1, numbers all of whose prime divisors are not of the form 4u+1. - Antti Karttunen, Dec 26 2020

			3,4,5; number 5 is not in this sequence.
5,12,13; number 13 is not in this sequence.
8,15,17; number 17 is not in this sequence.
7,24,25; number 25 is not in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A004613, A008846, A020882, A024362, A065338, A137407.

Subsequences: A125667 (the odd terms), A339875.

import Data.List (elemIndices)
a137409 n = a137409_list !! (n-1)
a137409_list = map (+ 1) $ elemIndices 0 a024362_list
-- Reinhard Zumkeller, Dec 02 2012

okQ[1] = True;
okQ[n_] := AnyTrue[FactorInteger[n][[All, 1]], Mod[#, 4] != 1&];
Select[Range[100], okQ] (* Jean-François Alcover, Mar 10 2019, after Federico Provvedi's comment *)

A065338(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = (f[i, 1]%4)); factorback(f); };
isA137409(n) = ((1==n)||(A065338(n)>1)); \\ Antti Karttunen, Dec 26 2020

Extended by R. J. Mathar, Aug 15 2010

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

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

Henry Bottomley, Jul 26 2001

nonn

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

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

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

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 *)

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

cp's OEIS Frontend

A024409 Hypotenuses of more than one primitive Pythagorean triangle.

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A014498 Varying radii of inscribed circles within primitive Pythagorean triples as a function of increasing values of hypotenuse.

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

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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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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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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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A137409 Numbers that cannot be the value of 'C' in a primitive Pythagorean triple (A < B; A^2 + B^2 = C^2).

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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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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.