A020886 -id:A020886 - cp's OEIS frontend

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.

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

A343894 Perimeters 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. The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide, in increasing order of side b.

Original entry on oeis.org

13, 37, 47, 71, 73, 107, 121, 143, 183, 177, 181, 191, 241, 239, 249, 253, 291, 299, 347, 337, 359, 409, 421, 429, 431, 433, 491, 517, 503, 529, 563, 537, 541, 579, 587, 649, 659, 661, 671, 753, 743, 769, 759, 781, 831, 767, 789, 793, 897, 851, 923, 863, 913, 947, 1033, 933

Offset: 1

Bernard Schott, May 07 2021

nonn

The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide then in increasing order of side b.
The sequence is not monotonic: a(9) = 183 > a(10) = 177.
All terms are odd.
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(3) = 15 + 12 + 20 = 47, because the third triple is (15, 12, 20) with relations 2/15 = 1/12 + 1/20 and 20-15 < 12 < 20+15.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343895.

Cf. A020886, A020890.

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

PARI

lista(nn) = {my(list = List()); for (a=4, nn, for (b = floor(a/2)+1, a-1, my(c = a*b/(2*b-a)); if ((denominator(c) == 1) && (gcd([a, b, c]) == 1) && (c-bMichel Marcus, May 10 2021

a(n) = A343891(n, 1) + A343891(n, 2) + A343891(n, 3).

a(n) = A020883(n) + A343892(n) + A343893(n).

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

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).

A078926 Number of primitive Pythagorean triangles with perimeter 2n.

Original entry on oeis.org

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

Offset: 1

Dean Hickerson, Dec 15 2002

A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.

Equivalently, number of odd unitary divisors d of n such that sqrt(n) < d < sqrt(2n). (A divisor d of n is 'unitary' if gcd(d,n/d) = 1.) Sketch of proof: A primitive Pythagorean triangle has edge lengths (r^2-s^2, 2rs, r^2+s^2), where 1<=s

			a(858)=2; the primitive Pythagorean triangles with edge lengths (364, 627, 725) and (195, 748, 773) both have perimeter 2*858 = 1716.

Antti Karttunen, Table of n, a(n) for n = 1..158730

a(n) = A070109(2n). A078927(n) is smallest s such that a(s)=n. a(n) is nonzero iff n is in A020886.

UnitaryDivisors :=
  func;
A078926:=
  func;
[A078926(n):n in [1..105]];

oddpart[n_] := If[OddQ[n], n, oddpart[n/2]]; a[n_] := Length[Select[Divisors[oddpart[n]], n<#^2<2n&&GCD[ #, n/# ]==1&]]
(* Second program: *)
Table[DivisorSum[n/2^IntegerExponent[n, 2], 1 &, n < #^2 < 2 n && CoprimeQ[#, n/#] &], {n, 105}] (* Michael De Vlieger, Oct 08 2017 *)

A078926(n) = sumdiv(n,d,(d%2)*(1==gcd(d,n/d))*((d*d)>n)*((d*d)<(2*n))); \\ Antti Karttunen, Oct 07 2017

Secondary offset added by Antti Karttunen, Oct 07 2017

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

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

A081804 Hypotenuses of primitive Pythagorean triangles sorted on semiperimeter.

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 65, 73, 85, 85, 89, 101, 97, 113, 109, 125, 145, 145, 137, 149, 157, 181, 173, 169, 185, 197, 185, 193, 221, 205, 205, 229, 221, 257, 233, 265, 241, 269, 265, 277, 293, 313, 281, 325, 289, 317, 305, 305, 365, 325, 365, 337

Offset: 1

Lekraj Beedassy, Apr 23 2003

nonn

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

Cf. A020886, A020882.

Corrected and extended by Ray Chandler, Oct 29 2003

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

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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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A343894 Perimeters 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. The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide, in increasing order of side b.

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Maple

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A063011 Ordered products of the sides of primitive Pythagorean triangles.

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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.

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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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Maple

Formula

A078926 Number of primitive Pythagorean triangles with perimeter 2n.

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Magma

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Extensions

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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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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A081804 Hypotenuses of primitive Pythagorean triangles sorted on semiperimeter.

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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=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.

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.