A057096 -id:A057096 - cp's OEIS frontend

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

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

Henry Bottomley, Aug 01 2000

nonn

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

Cf. A009000, A057096.

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 *)

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

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 *)

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

Henry Bottomley, Aug 01 2000

nonn

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

Cf. A009004, A009005, A046083, A057096.

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 *)

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

A057099 Middle side of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

4, 8, 12, 12, 15, 16, 24, 24, 20, 21, 24, 40, 35, 30, 28, 36, 32, 48, 60, 36, 48, 45, 40, 63, 45, 44, 84, 42, 60, 48, 72, 80, 56, 70, 60, 52, 56, 72, 112, 55, 99, 60, 77, 64, 75, 84, 96, 80, 68, 120, 63, 72, 144, 120, 96, 76, 105, 90, 72, 80, 143, 126, 120, 90, 84, 108, 91

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

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

Cf. A009012, A009023, A046084, A057096.

maxShortLeg = 66; terms = 67;
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, 2]] (* Jean-François Alcover, Nov 21 2019 *)

a(n) =A057096(n)/(A057098(n)*A057100(n)) =sqrt(A057100(n)^2-A057098(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

Henry Bottomley, Aug 01 2000

nonn

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

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

Vincenzo Librandi, Table of n, a(n) for n = 1..386

Cf. A000578 (cubes).

(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 *)

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

A081779 Ordered m for which m = k^3ab*(a^4 - b^4) determine (unique) solution triples(k,a,b), where k=1,2,3,... and (a,b) are coprime pairs, not both odd (i.e., of opposite parity).

Original entry on oeis.org

30, 240, 390, 810, 1020, 1920, 2100, 3120, 3750, 6090, 6480, 7380, 7770, 8160, 10290, 10530, 15360, 16800, 20130, 21870, 24960, 27540, 30000, 32760, 33390, 39930, 46410, 48720, 48750, 51840, 56700, 59040, 60060, 62160, 65280, 65910, 82320, 84240, 94920, 96360

Offset: 1

Lekraj Beedassy, Apr 10 2003

Also called Saint-Exupery numbers.

N. des Vallieres and F. D'Agay, Le Probleme du Pharaon.

Cf. A057096.

Import["https://oeis.org/A057096/b057096.txt", "Table"][[;;40, 2]]/2 (* Jean-François Alcover, Nov 21 2019 *)

a(n) = A057096(n)/2.

Corrected and extended by Jean-François Alcover, Nov 21 2019

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

Henry Bottomley, Aug 01 2000

nonn

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

Cf. A009111, A009112, A057096.

a(n) =A057096(n)/(2*A057100(n)) =A057098(n)*A057099(n)/2

cp's OEIS Frontend

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

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A063011 Ordered products of the sides of primitive Pythagorean triangles.

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

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A057099 Middle side of a Pythagorean triangle (ordered by the product of the sides).

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

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A081779 Ordered m for which m = k^3*a*b*(a^4 - b^4) determine (unique) solution triples(k,a,b), where k=1,2,3,... and (a,b) are coprime pairs, not both odd (i.e., of opposite parity).

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Comments

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Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Formula

A081779 Ordered m for which m = k^3ab*(a^4 - b^4) determine (unique) solution triples(k,a,b), where k=1,2,3,... and (a,b) are coprime pairs, not both odd (i.e., of opposite parity).