A024406 -id:A024406 - cp's OEIS frontend

A121727 Hypotenuse of primitive Pythagorean triangles sorted on area (A024406), then on hypotenuse.

5, 13, 17, 25, 41, 29, 37, 61, 65, 85, 53, 113, 65, 101, 145, 73, 85, 89, 181, 145, 221, 97, 125, 109, 197, 265, 149, 313, 257, 173, 137, 365, 185, 157, 325, 421, 229, 169, 481, 205, 185, 193, 401, 269, 545, 293

Offset: 1

Lekraj Beedassy, Aug 17 2006

nonn

Complete triple (X,Y,Z), with X>Y>Z is given by X=a(n),Y=A121728(n),Z=A121729(n).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A024406, A121728, A121729.

N:= 100000: # for triples with area <= N
R:= NULL:
for n from 1 while (2*n+1)*(n+1)*n <= N do
  for m from n+1 by 2 while (m^2 - n^2)*m*n <= N do
    if igcd(m,n) = 1 then
      R:= R, [m^2-n^2,2*m*n,m^2+n^2,(m^2-n^2)*m*n]
    fi
od od:
R:= sort([R], (s,t) -> s[4] < t[4] or (s[4] = t[4] and s[3] < t[3])):
R[..,3]; # Robert Israel, Dec 30 2024

v=vector(M=10^4); for(a=1, M, v[a] = []; fordiv(2*a, x, if(x<(y=2*a/x) && issquare(x^2+y^2, &z) && 1==gcd([x,y,z]), v[a] = concat(z, v[a])))); concat(v) /* Michael Somos, Dec 21 2016 */

a(20)=145 corrected by Philippe Guglielmetti, Dec 14 2016

a(43)=401 inserted by Michael Somos, Dec 21 2016

A094182 Inradii of primitive Pythagorean triangles, sorted on area A024406(n), then on inradii.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 5, 7, 6, 10, 7, 12, 9, 8, 15, 14, 15, 9, 11, 10, 20, 18, 13, 21, 11, 21, 12, 15, 22, 28, 13, 24, 30, 17, 14, 26, 35, 15, 33, 36, 19, 35, 30, 16, 30, 42, 21, 17, 40, 33, 39, 45, 44, 18, 34, 23, 45, 19, 20, 25, 39, 38, 55, 52, 56, 21, 27, 42, 22, 63, 51, 55, 42

Offset: 1

Lekraj Beedassy, May 06 2004

nonn

Corrected and extended by Ray Chandler, Jun 19 2004

A094183 Semiperimeter of primitive Pythagorean triangles, sorted on area A024406(n), then on semiperimeter.

Original entry on oeis.org

6, 15, 20, 28, 45, 35, 42, 66, 72, 91, 63, 120, 77, 110, 153, 88, 99, 104, 190, 156, 231, 117, 143, 130, 210, 276, 170, 325, 272, 195, 165, 378, 209, 187, 342, 435, 255, 204, 496, 238, 221, 228, 420, 299, 561, 323, 247, 506, 630, 273, 350, 304, 266, 285, 703

Offset: 1

Lekraj Beedassy, May 06 2004

nonn

Corrected and extended by Ray Chandler, Jun 19 2004

A024365 Areas of right triangles with coprime integer sides.

Original entry on oeis.org

6, 30, 60, 84, 180, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 8970, 8976, 9690, 10374

Offset: 1

David W. Wilson

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives areas A*B/2.
By Theorem 2 of Mohanty and Mohanty, all these numbers are primitive Pythagorean. - T. D. Noe, Sep 24 2013
This sequence also gives Fibonacci's congruous numbers (without multiplicity, in increasing order) divided by 4. See A258150. - Wolfdieter Lang, Jun 14 2015
The same as A024406 with duplicates removed. All terms are multiples of 6, cf. A258151. - M. F. Hasler, Jan 20 2019

			6 is in the sequence because it is the area of the 3-4-5 triangle.
a(7) = 210 corresponds to the two primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). See A024406. - _Wolfdieter Lang_, Jun 14 2015

T. D. Noe, Table of n, a(n) for n = 1..10000 (corrected by _Giovanni Resta_, Jan 21 2019)
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.

Cf. A009111, A009112, A024406 (with multiplicity), A258150, A024407, A258151 (terms divided by 6).

Subsequence of A073120 and A147778.

nn = 22; (* nn must be even *) t = Union[Flatten[Table[If[GCD[u, v] == 1 && Mod[u, 2] + Mod[v, 2] == 1, u v (u^2 - v^2), 0], {u, nn}, {v, u - 1}]]]; Select[Rest[t], # < nn (nn^2 - 1) &] (* T. D. Noe, Sep 19 2013 *)

select( {is_A024365(n)=my(N=1+#n=divisors(2*n)); for(i=1, N\2, gcd(n[i], n[N-i])==1 && issquare(n[i]^2+n[N-i]^2) && return(n[i]))}, [1..10^4]) \\ is_A024365 returns the smaller leg if n is a term, else 0. - M. F. Hasler, Jun 06 2024

Positive integers of the form u*v*(u^2 - v^2) where 2uv and u^2 - v^2 are coprime or, alternatively, where u, v are coprime and one of them is even.

a(n) = 6*A258151(n). - M. F. Hasler, Jan 20 2019

Additional comments James R. Buddenhagen, Aug 10 2008 and from Max Alekseyev, Nov 12 2008

Edited by N. J. A. Sloane, Nov 20 2008 at the suggestion of R. J. Mathar

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

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

A024407 Areas of more than one primitive Pythagorean triangle.

Original entry on oeis.org

210, 2730, 7980, 71610, 85470, 106260, 114114, 234780, 341880, 420420, 499590, 1563660, 1647030, 1857240, 2042040, 3423420, 3666390, 6587490, 7393470, 8514660, 9279270, 12766110, 13123110, 17957940, 18820830, 23393370, 23573550, 29099070, 29274630, 29609580

Offset: 1

David W. Wilson

nonn

Among a(1) to a(30), only a(23) = 13123110 has multiplicity 3, the others have multiplicity 2. The three primitive Pythagorean triangles corresponding to a(23) are [4485, 5852, 7373], [3059, 8580, 9109] and [19019, 1380, 19069]. Leg exchange is not taken into account. - Wolfdieter Lang, Jun 15 2015

The area 13123110 of multiplicity three was discovered by C. L. Shedd in 1945, cf. Beiler, Gardner and Weisstein. - M. F. Hasler, Jan 20 2019

			The first repeated terms in A024406 are:
   A024406(6) = A024406(7) = 210 = a(1),
   A024406(24) = A024406(25) = 2730 = a(2),
   A024406(42) = A024406(43) = 7980 = a(3). - _M. F. Hasler_, Jan 20 2019

A. H. Beiler: The Eternal Triangle. Ch. 14 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, 1966, p. 127.
M. Gardner: The Sixth Book of Mathematical Games from Scientific American. University of Chicago Press, 1984, pp. 160-161.

Giovanni Resta, Table of n, a(n) for n = 1..200
Ron Knott, Pythagorean Triples and Online Calculators
Eric W. Weisstein, Primitive Right Triangle, on MathWorld.Wolfram.com.

Cf. A024365, A024406.

Terms occurring more than once in A024406 listed exactly once: { n = A024406(k): n = A024406(k+m), m > 0 }. - M. F. Hasler, Jan 20 2019, edited by David A. Corneth, Jan 21 2019

a(29) and a(30) added by Wolfdieter Lang, Jun 14 2015

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

A121727 Hypotenuse of primitive Pythagorean triangles sorted on area (A024406), then on hypotenuse.

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A094182 Inradii of primitive Pythagorean triangles, sorted on area A024406(n), then on inradii.

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A094183 Semiperimeter of primitive Pythagorean triangles, sorted on area A024406(n), then on semiperimeter.

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A024365 Areas of right triangles with coprime integer sides.

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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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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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A024407 Areas of more than one primitive Pythagorean triangle.

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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.