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.

Showing 1-5 of 5 results.

A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.

Original entry on oeis.org

60, 480, 780, 1620, 2040, 3840, 4200, 6240, 7500, 12180, 12960, 14760, 15540, 16320, 20580, 21060, 30720, 33600, 40260, 43740, 49920, 55080, 60000, 65520, 66780, 79860, 92820, 97440, 97500, 103680, 113400, 118080, 120120, 124320, 130560, 131820, 164640
Offset: 1

Views

Author

Henry Bottomley, Aug 01 2000

Keywords

Comments

It is an open question whether any two distinct Pythagorean Triples can have the same product of their sides.
From Amiram Eldar, Nov 22 2020: (Start)
Named after the French writer Antoine de Saint-Exupéry (1900-1944).
The problem of finding two distinct Pythagorean triples with the same product was proposed by Eckert (1984). It is equivalent of finding a nontrivial solution of the Diophantine equation x*y*(x^4-y^4) = z*w*(z^4-w^4) (Bremner and Guy, 1988). (End)

Examples

			a(1) = 3*4*5 = 60.

References

  • Richard K. Guy, "Triangles with Integer Sides, Medians and Area." D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.
  • Antoine de Saint-Exupéry, Problème du Pharaon, Liège : Editions Dynamo, 1957.

Links

Crossrefs

Cf. A009004, A009012, A009111, A020886, A057097, A057098, A057099, A057100.

Programs

  • Mathematica
    k=5000000; lst={}; Do[Do[If[IntegerQ[a=Sqrt[c^2-b^2]], If[a>=b, Break[]]; x=a*b*c; If[x<=k, AppendTo[lst,x]]], {b,c-1,4,-1}], {c,5,400,1}]; Union@lst (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009 *)

Formula

a(n) = 60*A057097(n) = A057098(n)*A057099(n)*A057100(n).

A057100 Hypotenuses of Pythagorean triangles (ordered by the product of the sides).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 25, 29, 30, 41, 37, 34, 35, 39, 40, 50, 61, 45, 52, 51, 50, 65, 53, 55, 85, 58, 65, 60, 75, 82, 65, 74, 68, 65, 70, 78, 113, 73, 101, 75, 85, 80, 85, 91, 100, 89, 85, 122, 87, 90, 145, 123, 104, 95, 111, 102, 97, 100, 145, 130, 125, 106
Offset: 1

Views

Author

Henry Bottomley, Aug 01 2000

Keywords

Examples

			a(1)=5 since 3*4*5=60 is smallest possible positive product

Crossrefs

Cf. A009000, A057096.

Programs

  • Mathematica
    maxShortLeg = 66; terms = 64;
r[a_] := {a, b, c} /. {ToRules[Reduce[a <= b < c && a^2+b^2 == c^2, {b, c}, Integers]]};
abc = r /@ Complement[Range[maxShortLeg], {1, 2, 4}] // Flatten[#, 1]&;
SortBy[abc, Times @@ # &][[;; terms, 3]] (* Jean-François Alcover, Nov 21 2019 *)

Formula

a(n) =A057096(n)/(A057098(n)*A057099(n)) =sqrt(A057098(n)^2+A057099(n)^2)

A057098 Shortest side of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 7, 10, 15, 20, 18, 9, 12, 16, 21, 15, 24, 14, 11, 27, 20, 24, 30, 16, 28, 33, 13, 40, 25, 36, 21, 18, 33, 24, 32, 39, 42, 30, 15, 48, 20, 45, 36, 48, 40, 35, 28, 39, 51, 22, 60, 54, 17, 27, 40, 57, 36, 48, 65, 60, 24, 32, 35, 56, 63, 45, 60, 19, 66, 44, 56
Offset: 1

Views

Author

Henry Bottomley, Aug 01 2000

Keywords

Examples

			a(1)=3 since 3*4*5=60 is smallest possible positive product

Crossrefs

Cf. A009004, A009005, A046083, A057096.

Programs

  • Mathematica
    maxShortLeg = 66; terms = 71;
r[a_] := {a, b, c} /. {ToRules[Reduce[a <= b < c && a^2+b^2 == c^2, {b, c}, Integers]]};
abc = r /@ Complement[Range[maxShortLeg], {1, 2, 4}] // Flatten[#, 1]&;
SortBy[abc, Times @@ # &][[;; terms, 1]] (* Jean-François Alcover, Nov 21 2019 *)

Formula

a(n) =A057096(n)/(A057099(n)*A057100(n)) =sqrt(A057100(n)^2-A057099(n)^2)

A057097 Products of the three sides of Pythagorean triangles divided by 60.

Original entry on oeis.org

1, 8, 13, 27, 34, 64, 70, 104, 125, 203, 216, 246, 259, 272, 343, 351, 512, 560, 671, 729, 832, 918, 1000, 1092, 1113, 1331, 1547, 1624, 1625, 1728, 1890, 1968, 2002, 2072, 2176, 2197, 2744, 2808, 3164, 3212, 3333, 3375, 3927, 4096, 4250, 4459, 4480
Offset: 1

Views

Author

Henry Bottomley, Aug 01 2000

Keywords

Comments

Note that if m appears in the sequence then k^3*m will also appear for all k and so in particular all cubes appear; the reverse is not always true (for example, 32*255*257/60 = 34952 = 2^3*4369 eventually appears, but 4369 does not).
By considering the Pythagorean triangle (3k, 4k, 5k) we see that all numbers k^3 are in the sequence. - Sergey Pavlov, Mar 29 2017

Examples

			a(1) = 3*4*5/60 = 1.

Links

Crossrefs

Cf. A000578 (cubes).

Programs

  • Mathematica
    (k=600000; lst={}; Do[Do[If[IntegerQ[a=Sqrt[c^2 - b^2]], If[a>=b, Break[]]; x=a b c; If[x<=k, AppendTo[lst, x]]], {b, c - 1, 4, -1}], {c, 5, 400, 1}]; Union@lst)/60 (* Vincenzo Librandi Mar 30 2017 *)

Formula

a(n) = A057096(n)/60 = A057098(n)*A057099(n)*A057100(n)/60.

A057101 Area of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

6, 24, 30, 54, 60, 96, 84, 120, 150, 210, 216, 180, 210, 240, 294, 270, 384, 336, 330, 486, 480, 540, 600, 504, 630, 726, 546, 840, 750, 864, 756, 720, 924, 840, 960, 1014, 1176, 1080, 840, 1320, 990, 1350, 1386, 1536, 1500, 1470, 1344, 1560, 1734, 1320
Offset: 1

Views

Author

Henry Bottomley, Aug 01 2000

Keywords

Examples

			a(1)=3*4/2=6 since 3*4*5=60 is smallest possible positive product

Crossrefs

Cf. A009111, A009112, A057096.

Formula

a(n) =A057096(n)/(2*A057100(n)) =A057098(n)*A057099(n)/2
Showing 1-5 of 5 results.

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