A020883 -id:A020883 - cp's OEIS frontend

A343892 Side b of integer-sided primitive triangles (a, b, c) 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.

3, 10, 12, 15, 21, 30, 36, 35, 40, 44, 55, 56, 52, 63, 65, 78, 70, 90, 77, 105, 99, 85, 102, 119, 132, 136, 117, 114, 143, 133, 126, 152, 171, 154, 182, 168, 165, 210, 195, 161, 176, 184, 208, 207, 187, 240, 230, 253, 200, 221, 198, 255, 225, 234, 216, 275, 300, 306, 247, 270

Offset: 1

Bernard Schott, May 06 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
The sequence is not increasing because a(7) = 36 > a(8) = 35, but, these sides b are listed in increasing order in A020890.
The first term appearing twice is 330 and corresponds to triples (435, 330, 638) and (460, 330, 759), the second one is 462 and corresponds to triples (483, 462, 506) and (532, 462, 627).
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(4) = 15, because the fourth triple is (21, 15, 35) with side b = 15, satisfying 1/15 = 2/21 - 1/35 and 31-15 < 21 < 31+15.

Cf. A343891 (triples), A020883 (side a), A343893 (side c), A343894 (perimeter).

Cf. A020890 (sides b ordered).

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b

a(n) = A343891(n, 2).

A343893 Side c of integer-sided primitive triangles (a, b, c) 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.

Original entry on oeis.org

6, 15, 20, 35, 28, 42, 45, 63, 88, 77, 66, 72, 117, 99, 104, 91, 130, 110, 165, 120, 143, 204, 187, 170, 156, 153, 221, 247, 195, 228, 266, 209, 190, 238, 210, 273, 285, 231, 255, 368, 336, 345, 304, 322, 391, 272, 299, 276, 425, 357, 450, 323, 400, 414, 513, 350, 325, 342, 475, 459

Offset: 1

Bernard Schott, May 06 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
The sequence is not increasing because a(4) = 35 > a(5) =  28, but, these sides c are listed in increasing order in A020886.
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(3) = 20, because the third triple is (15, 12, 20) with side c = 20, satisfying 1/20 = 2/15 - 1/12 and 15-12 < 20 < 15+12.

Cf. A343891 (triples), A020883 (side a), A343892 (side b), A343894 (perimeter).

Cf. A020886 (sides c ordered).

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b

a(n) = A343891(n, 3).

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

A020890 Ordered set of (b + c - a)/2 as (a,b,c) runs through all primitive Pythagorean triples with a < b < c.

Original entry on oeis.org

3, 10, 12, 15, 21, 30, 35, 36, 40, 44, 52, 55, 56, 63, 65, 70, 77, 78, 85, 90, 99, 102, 105, 114, 117, 119, 126, 132, 133, 136, 143, 152, 154, 161, 165, 168, 171, 176, 182, 184, 187, 195, 198, 200, 207, 208, 210, 216, 221, 225, 230, 234, 240, 247, 253, 255, 260, 261, 270

Offset: 1

Clark Kimberling

nonn

From Bernard Schott, May 06 2021: (Start)
Also, ordered sides b 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 (A343892).
The first term appearing twice is 330 = a(71) = a(72). (End)

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

Cf. A020889.

Cf. Triangles with 2/a = 1/b + 1/c: A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

a(n) = A020889(n)/2.

Offset corrected to 1 by Ray Chandler, Jan 23 2020

A081934 Ordered odd long legs of primitive Pythagorean triangles.

Original entry on oeis.org

15, 21, 35, 45, 55, 63, 77, 91, 99, 105, 117, 143, 153, 165, 171, 187, 195, 209, 221, 231, 247, 253, 255, 273, 275, 285, 299, 323, 325, 345, 351, 357, 377, 391, 399, 403, 425, 435, 437, 459, 465, 475, 483, 493, 513, 525, 527, 551, 561, 575, 589, 595, 609, 621

Offset: 1

Lekraj Beedassy, Apr 23 2003

nonn

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

Cf. A020883.

More terms from Ray Chandler, Oct 29 2003

A155177 Area ar/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, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

1, 5, 140, 385, 2870, 8555, 29370, 42925, 93665, 116795, 149226, 155155, 348551, 380380, 414090, 513590, 1229305, 1801800, 2567895, 2767905, 3873301, 3924830, 5053620, 6970150, 17090486, 18362930, 23396450, 31919165, 39336465, 41791750

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5, ar=3*4/2=6, s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176

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

A381006 Ordered long legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

24, 288, 4224, 66048, 1050624, 16785408, 268468224, 4295098368, 68720001024, 1099513724928, 17592194433024, 281475010265088, 4503599761588224, 72057594574798848, 1152921506754330624, 18446744082299486208, 295147905213712564224, 4722366483007084167168

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A020883.
Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A020883.

Cf. A381005 (short legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).

Magma
```
[2^(4*n) + 2^(2*n+1): n in [1..20]];
```

A381006[n_] := #*(# + 2) & [4^n]; Array[A381006, 20] (* or *)
LinearRecurrence[{20, -64}, {24, 288}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI
```
a(n) = 2^(4*n) + 2^(2*n+1)
```

def A381006(n): return (m:=1<<(n<<1))*(m+2) # Chai Wah Wu, Feb 13 2025

a(n) = 2^(4n) + 2^(2n+1).
a(n) = sqrt( A381007(n)^2 - A381005(n)^2 ).
G.f.: 24*(1 - 8*x)/((1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A024360 Number of primitive Pythagorean triangles with long leg n.

Original entry on oeis.org

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

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 B takes value n.

Number of times n occurs in A020883.

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

Cf. A020883, A024359, A024361.

A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[;; 10000, 2]]];
A@024361 - A@024359 (* Jean-François Alcover, Mar 27 2020 *)

a(n) = A024361(n) - A024359(n). - Ray Chandler, Feb 03 2020

A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

Original entry on oeis.org

1, 7916, 35882, 37816, 47491, 128429, 131830, 146471, 154799, 157579, 170219, 174964, 187544, 207829, 208039, 222887, 223142, 262502, 291544, 319825, 327602, 331627, 353857, 476681, 477659, 494207, 522025, 537454, 540682, 558161, 571670

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes,pr=3*4*5=60-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;pr=a*b*c;If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[pr-1]&&PrimeQ[pr+1],AppendTo[lst,n]],{n,3*9!}];lst

A024355 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of AUB, sorted.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 11, 12, 12, 13, 15, 15, 16, 17, 19, 20, 20, 21, 21, 23, 24, 24, 25, 27, 28, 28, 29, 31, 32, 33, 33, 35, 35, 36, 36, 37, 39, 39, 40, 40, 41, 43, 44, 44, 45, 45, 47, 48, 48, 49, 51, 51, 52, 52, 53, 55, 55, 56, 56, 57, 57, 59, 60, 60, 60, 60, 61, 63, 63, 64, 65

Offset: 1

David W. Wilson

nonn

Union of A020883 and A020884, sorted (with multiplicity); n occurs A024361(n) times. - Ray Chandler, Feb 03 2020

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

Cf. A020883, A020884, A024361.

cp's OEIS Frontend

A343892 Side b of integer-sided primitive triangles (a, b, c) 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.

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A343893 Side c of integer-sided primitive triangles (a, b, c) 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.

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

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A020890 Ordered set of (b + c - a)/2 as (a,b,c) runs through all primitive Pythagorean triples with a < b < c.

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A081934 Ordered odd long legs of primitive Pythagorean triangles.

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A155177 Area ar/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=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes.

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A381006 Ordered long legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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PARI

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Formula

A024360 Number of primitive Pythagorean triangles with long leg n.

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A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes, pr=a*b*c, pr-+1 are primes.

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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.

A155177 Area ar/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, ar=a*b/2; s=a+b+c, s-+1 are primes.

A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.