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A195770 Positive integers a for which there is a 1-Pythagorean triple (a,b,c) satisfying a<=b. 217
3, 5, 6, 7, 7, 9, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32, 32, 33, 33, 33, 33 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

In case the number k=-cos(C) is a rational number, the law of cosines, c^2=a^2+b^2+k*a*b, can be regarded as a Diophantine equation having positive integer solutions a,b,c satisfying a<=b.  The terms "k-Pythagorean triple" and "primitive k-Pythagorean triple" generalize the classical terms corresponding to the case k=0.

Example: the first five (3/2)-Pythagorean triples are

(5,18,22),(6,11,16),(9,11,71),(10,36,44),(12,22,32);

the first five primitive (3/2)-Pythagorean triples are

(5,18,22),(6,11,16),(9,64,71),(13,138,148),(14,75,86).

...

If |k|>2, there is no triangle with sidelengths a,b,c satisfying c^2=a^2+b^2+k*a*b, but this equation is, nevertheless, a Diophantine equation for rational k.

  ...

Related sequences (k-Pythagorean triples):

k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)

0.......(3,4,5).............A009004..A156681..A156682

1.......(3,5,7).............A195770..A195866..A195867

3.......(3,7,11)............A196112..A196113..A196114

4.......(3,8,13)............A196119..A196120..A196121

5.......(1,3,5).............A196155..A196156..A196157

6.......(2,3,7).............A196162..A196163..A196164

7.......(1,1,3).............A196169..A196170..A196171

8.......(1,4,7).............A196176..A196177..A196178

9.......(1,15,19)...........A196183..A196184..A196185

10......(1,2,5).............A196238..A196239..A196240

1/2.....(2,3,4).............A195879..A195880..A195881

3/2.....(5,18,22)...........A195925..A195926..A195927

1/3.....(3,8,9).............A195939..A195940..A195941

2/3.....(4,9,11)............A196001..A196002..A196003

4/3.....(7,36,41)...........A196040..A196041..A196042

5/3.....(7,39,45)...........A196088..A196089..A196090

5/2.....(5,22,28)...........A196026..A196027..A196028

1/4.....(2,2,3).............A196259..A196260..A196261

3/4.....(2,6,7).............A196252..A196253..A196254

5/4.....(3,20,22)...........A196098..A196099..A196100

7/4.....(9,68,76)...........A196105..A196106..A196107

1/5.....(5,7,9).............A196348..A196349..A196350

1/8.....(4,10,11)...........A196355..A196356..A196357

-1......(1,1,1).............A195778..A195794..A195795

-3......(1,3,1).............A196369..A196370..A196371

-4......(1,4,1..............A196376..A196377..A196378

-5......(1,5,1).............A196383..A196384..A196385

-6......(1,6,1).............A196390..A196391..A196392

-1/2....(1,2,2).............A195872..A195873..A195874

-3/2....(2,3,2).............A195918..A195919..A195920

-5/2....(2,5,2).............A196362..A196363..A196364

-1/3....(1,3,3).............A195932..A195933..A195934

-2/3....(2,3,3).............A195994..A195995..A195996

-4/3....(3,4,3).............A196033..A196034..A196035

-5/3....(3,5,3).............A196008..A196009..A196083

-1/4....(1,4,4).............A196266..A196267..A196268

-3/4....(3,4,4).............A196245..A196247..A196248

...

Related sequences (primitive k-Pythagorean triples):

k...(a(1),b(1),c(1))........a(n).....b(n).....c(n)

0.......(3,4,5).............A020884..A156678..A156679

1.......(3,5,7).............A195868..A195869..A195870

3.......(3,7,11)............A196115..A196116..A196117

4.......(3,8,13)............A196122..A196123..A196124

5.......(1,3,5).............A196158..A196159..A196160

6.......(2,3,7).............A196165..A196166..A196167

7.......(1,1,3).............A196172..A196173..A196174

8.......(1,4,7).............A196179..A196180..A196181

9.......(1,15,19)...........A196186..A196187..A196188

10......(1,2,5).............A196241..A196242..A196243

1/2.....(2,3,4).............A195882..A195883..A195884

3/2.....(5,18,22)...........A195928..A195929..A195930

1/3.....(3,8,9).............A195990..A195991..A195992

2/3.....(4,9,11)............A196004..A196005..A196006

4/3.....(7,36,41)...........A196043..A196044..A196045

5/3.....(7,39,45)...........A196091..A196092..A196093

5/2.....(5,22,28)...........A196029..A196030..A196031

1/4.....(2,2,3).............A196262..A196263..A196264

3/4.....(2,6,7).............A196255..A196256..A196257

5/4.....(3,20,22)...........A196101..A196102..A196103

7/4.....(9,68,76)...........A196108..A196109..A196110

1/5.....(5,7,9).............A196351..A196352..A196353

1/8.....(4,10,11)...........A196358..A196359..A196360

-1......(1,1,1))............A195796..A195862..A195863

-3......(1,3,1).............A196372..A196373..A196374

-4......(1,4,1..............A196379..A196380..A196381

-5......(1,5,1).............A196386..A196387..A196388

-6......(1,6,1).............A196393..A196394..A196395

-1/2....(1,2,2).............A195875..A195876..A195877

-3/2....(2,3,2).............A195921..A195922..A195923

-5/2....(2,5,2).............A196365..A196366..A196367

-1/3....(1,3,3).............A195935..A195936..A195937

-2/3....(2,3,3).............A195997..A195998..A195999

-4/3....(3,4,3).............A196036..A196037..A196038

-5/3....(3,5,3).............A196084..A196085..A196086

-1/4....(1,4,4).............A196269..A196270..A196271

-3/4....(3,4,4).............A196249..A196250..A196246

From Georg Fischer, Oct 26 2020: (Start)

The Mathematica program below has fixed limits (z7, z8, z9). Therefore, it misses higher values of b. For example, the following triples are do not show up in the corresponding sequences:

        A196112 A196113 A196114 - non-primitive 3-Pythagorean

    49:      29    1008    1051

        A196241 A196242 A196243 - primitive 10-Pythagorean

    31:      13     950    1013

This problem affects 62 of the 74 parameter combinations. (End)

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

The first seven 1-Pythagorean triples (a,b,c), ordered as

described above, are as follows:

3,5,7........7^2 = 3^2 + 5^2 + 3*5

5,16,19.....19^2 = 5^2 + 16^2 + 5*16

6,10,14.....14^2 = 6^2 + 10^2 + 6*10

7,8,13

7,33,37

9,15,21

9,56,61

10,32,38

MATHEMATICA

z8 = 2000; z9 = 400; z7 = 100;

k = 1; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];

d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]

t[a_] := Table[d[a, b], {b, a, z8}]

u[n_] := Delete[t[n], Position[t[n], 0]]

Table[u[n], {n, 1, 15}]

t = Table[u[n], {n, 1, z8}];

Flatten[Position[t, {}]]

u = Flatten[Delete[t, Position[t, {}]]];

x[n_] := u[[3 n - 2]];

Table[x[n], {n, 1, z7}]  (* A195770 *)

y[n_] := u[[3 n - 1]];

Table[y[n], {n, 1, z7}]  (* A195866 *)

z[n_] := u[[3 n]];

Table[z[n], {n, 1, z7}]  (* A195867 *)

x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]

y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]

z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]

f = Table[x1[n], {n, 1, z9}];

x2 = Delete[f, Position[f, 0]]  (* A195868 *)

g = Table[y1[n], {n, 1, z9}];

y2 = Delete[g, Position[g, 0]]  (* A195869 *)

h = Table[z1[n], {n, 1, z9}];

z2 = Delete[h, Position[h, 0]]  (* A195870 *)

CROSSREFS

Cf. A195866, A195867, A195868, A195869, A195870.

Sequence in context: A070083 A316851 A196778 * A196008 A004220 A202308

Adjacent sequences:  A195767 A195768 A195769 * A195771 A195772 A195773

KEYWORD

nonn

AUTHOR

Clark Kimberling, Sep 25 2011

STATUS

approved

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Last modified January 15 15:49 EST 2021. Contains 340187 sequences. (Running on oeis4.)