cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-40 of 53 results. Next

A024357 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of AUBUC, sorted.

Original entry on oeis.org

3, 4, 5, 5, 7, 8, 9, 11, 12, 12, 13, 13, 15, 15, 16, 17, 17, 19, 20, 20, 21, 21, 23, 24, 24, 25, 25, 27, 28, 28, 29, 29, 31, 32, 33, 33, 35, 35, 36, 36, 37, 37, 39, 39, 40, 40, 41, 41, 43, 44, 44, 45, 45, 47, 48, 48, 49, 51, 51, 52, 52, 53, 53, 55, 55, 56, 56, 57
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A020882, A020883, A020884, A042965.

A024410 Long leg of more than one primitive Pythagorean triangle.

Original entry on oeis.org

420, 572, 780, 840, 924, 1020, 1292, 1596, 1680, 1716, 1848, 1932, 2100, 2145, 2244, 2300, 2484, 2520, 2640, 2652, 2700, 2900, 2964, 3080, 3132, 3315, 3348, 3432, 3465, 3596, 3640, 3828, 3876, 3960, 4060, 4092, 4095, 4340, 4488, 4588, 4620, 4680, 4692
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Also, middle side a of more than one primitive integer-sided 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; hence, terms that appear more than once in A020883. - Bernard Schott, Oct 21 2021

Examples

			From _Bernard Schott_, Oct 21 2021: (Start)
-> For primitive Pythagorean triples:
a(1) = 420 because 420 is the smallest long leg that belongs to more than one primitive Pythagorean triples, we have 29^2 + 420^2 = 421^2 and 341^2 + 420^2 = 541^2.
-> For primitive triples with 2/a = 1/b + 1/c:
a(1) = 420 because 420 is the smallest middle side a that belongs to more than one primitive integer-sided 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, we have 2/420 = 1/310 + 1/651 and 2/420 = 1/406 + 1/435. (End)

Links

Crossrefs

Cf. A020883.

Programs

  • Mathematica
    bb=1;s=e="";For[b=1,b<=12^3,For[a=b-1,a>2,c=(a^2+b^2)^0.5;If[c==Round[c]&&GCD[a,b]==1,If[b==bb,e=e<>ToString[b]<>",";s=s<>ToString[a]<>","<>ToString[b]<>","<>ToString[Round[c]]<>"; "];bb=b];a-- ];b++ ];Print["B = ",e] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Tally[Union[Sort/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@(Select[ Subsets[Range[1,121,2],{2}],GCD@@#==1&]))][[All,2]]],#[[2]]>1&][[All,1]] //Sort (* Harvey P. Dale, Mar 07 2020 *)

A081872 Long legs of primitive Pythagorean triangles sorted on semiperimeter.

Original entry on oeis.org

4, 12, 15, 24, 21, 35, 40, 45, 60, 63, 56, 55, 84, 77, 80, 99, 72, 112, 91, 117, 144, 143, 105, 140, 132, 180, 165, 120, 176, 195, 153, 168, 220, 187, 156, 221, 171, 255, 208, 264, 209, 260, 247, 252, 285, 312, 231, 323, 240, 308, 273, 224, 364, 253, 357, 288
Offset: 1

Views

Author

Lekraj Beedassy, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A020886, A020883, A046087.

Extensions

More terms from Ray Chandler, Oct 29 2003

A081935 Ordered even long legs of primitive Pythagorean triangles.

Original entry on oeis.org

4, 12, 24, 40, 56, 60, 72, 80, 84, 112, 120, 132, 140, 144, 156, 168, 176, 180, 208, 220, 224, 240, 252, 260, 264, 272, 288, 304, 308, 312, 340, 352, 360, 364, 380, 396, 408, 416, 420, 420, 440, 456, 460, 468, 476, 480, 520, 528, 532, 544, 552, 572, 572, 600
Offset: 1

Views

Author

Lekraj Beedassy, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A020883.

Extensions

More terms from Ray Chandler, Oct 29 2003

A155180 Short leg A 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.

Original entry on oeis.org

3, 15833, 71765, 75633, 94983, 256859, 263661, 292943, 309599, 315159, 340439, 349929, 375089, 415659, 416079, 445775, 446285, 525005, 583089, 639651, 655205, 663255, 707715, 953363, 955319, 988415, 1044051, 1074909, 1081365, 1116323
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

Keywords

Comments

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

Crossrefs

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178

Programs

  • Mathematica
    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,a]],{n,3*9!}];lst

A165159 Long legs in primitive Pythagorean triangles with three side lengths of composite integers.

Original entry on oeis.org

56, 63, 77, 117, 120, 143, 153, 156, 171, 176, 187, 220, 224, 240, 247, 253, 273, 304, 323, 345, 352, 357, 360, 364, 377, 396, 403, 416, 435, 437, 456, 460, 468, 475, 476, 483, 493, 513, 525, 527, 528, 544, 561, 621, 624, 627, 644, 663, 665, 667, 672, 680
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Sep 06 2009

Keywords

Comments

The sequence collects the numbers B such that A^2+B^2=C^2, A
three of A, B and C are in A002808. If there are two or more triangles of this kind with the same B,
like (A,B,C) = (1003,1596,1885) and (A,B,C) = (1403,1596,2125), only one instance
of B is added to the sequence.

Examples

			(A,B,C)=(33,56,65) contributes B=56 to the sequence. (A,B,C)=(16,63,65) contributes B=63 to the sequence.

Crossrefs

Cf. A020882, A020883, A165158, A165160.

Programs

  • Mathematica
    lst={}; Do[Do[If[IntegerQ[c=Sqrt[a^2+b^2]] && GCD[a,b,c]==1,If[ !PrimeQ[a]&&!PrimeQ[b] && !PrimeQ[c], AppendTo[lst,b]]],{a,b-1,3,-1}], {b,4,2000,1}];Union@lst

Extensions

Edited by R. J. Mathar, Oct 02 2009

A165236 Short legs of primitive Pythagorean Triples (a,b,c) such that 2*a+1, 2*b+1 and 2*c+1 are primes.

Original entry on oeis.org

20, 33, 44, 56, 68, 273, 303, 320, 380, 440, 483, 740, 1071, 1089, 1101, 1220, 1376, 1484, 1635, 1773, 1808, 1869, 1940, 1965, 2000, 2120, 2144, 2204, 2319, 2715, 2763, 3003, 3164, 3309, 3500, 3603, 3729, 3740, 3753, 3801, 4148, 4215, 4323, 4340, 4401
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Sep 09 2009

Keywords

Comments

Only one instance of a enters the sequence if multiple solutions exist, like with (a,b,c) = (320,999,1049) and (a,b,c) = (320,25599,25601).
Subsequence of A009004. [R. J. Mathar, Mar 25 2010]

Examples

			(a,b,c) = (20,21,29), (33,56,65), (44,483,485), (56,783,785), (68,285,293), (273,4136,4145).
In the first case, for example, 2*20+1=41, 2*21+1 and 2*29+1 are all prime, which adds the half-leg 20 to the sequence.

Crossrefs

Cf. A009004, A020883, A165237, A165238

Programs

  • Mathematica
    amax=6*10^4;lst={};k=0;q=12!;Do[If[(e=((n+1)^2-n^2))>amax,Break[]];
Do[If[GCD[m, n]==1,a=m^2-n^2;If[PrimeQ[2*a+1],b=2*m*n;If[PrimeQ[2*b+1],If[GCD[a, b]==1,If[a>b,{a,b}={b,a}];If[a>amax,Break[]];
c=m^2+n^2;If[PrimeQ[2*c+1], k++;AppendTo[lst,a]]]]]];If[a>amax,Break[]],{m,n+1,12!,2}],{n,1,q, 1}];Union@lst

Extensions

Comments moved to examples and definition clarified by R. J. Mathar, Mar 25 2010

A165237 Long legs of primitive Pythagorean triples (a,b,c) for which 2a+1, 2b+1 and 2c+1 are primes.

Original entry on oeis.org

21, 56, 285, 483, 783, 999, 1269, 1593, 1911, 2613, 3003, 3596, 3621, 3740, 4136, 4233, 4928, 5096, 5451, 5828, 5840, 6320, 7040, 7280, 8036, 8468, 9021, 9296, 9591, 11660, 12075, 12573, 12705, 12920, 12956, 13563, 14396, 14595, 15429, 15561
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Sep 09 2009

Keywords

Examples

			See A165236.

Crossrefs

Cf. A020883, A165236, A156238.

Programs

  • Mathematica
    amax=6*10^4;lst={};k=0;q=12!;Do[If[(e=((n+1)^2-n^2))>amax,Break[]];Do[If[GCD[m,n]==1,a=m^2-n^2;If[PrimeQ[2*a+1],b=2*m*n;If[PrimeQ[2*b+1],If[GCD[a,b]==1,If[a>b,{a,b}={b,a}];If[a>amax,Break[]];c=m^2+n^2;If[PrimeQ[2*c+1],k++;AppendTo[lst,b]]]]]];If[a>amax,Break[]],{m,n+1,12!,2}],{n,1,q,1}];Union@lst

A376429 Numbers k that occur as longer legs of Pythagorean triangles with Pythagorean primes A002144 as hypotenuses.

Original entry on oeis.org

4, 12, 15, 21, 35, 40, 45, 55, 60, 72, 80, 91, 99, 105, 112, 132, 140, 165, 168, 180, 195, 208, 209, 221, 231, 252, 255, 260, 272, 275, 285, 288, 299, 308, 312, 325, 340, 351, 380, 391, 399, 408, 420, 425, 440, 459, 465, 520, 532, 551, 552, 572, 575, 589, 595, 600
Offset: 1

Views

Author

Hugo Pfoertner, Sep 22 2024

Keywords

Comments

Distinct sorted terms of A002365.

Crossrefs

Cf. A002144, A002365, A376428, A376430.
Subsequence of A020883.

A137407 Numbers that cannot be the length of the long leg in any primitive Pythagorean triple.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 87, 88
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 14 2008, Aug 28 2009

Keywords

Comments

Complement of A024354. [R. J. Mathar, Sep 03 2009]

Links

Crossrefs

Cf. A137409, A020883.

Extensions

Definition reworded by R. J. Mathar, Sep 03 2009
Previous Showing 31-40 of 53 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.