A195770 - OEIS

%I #38 Oct 24 2024 01:26:00

%S 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,

%T 20,21,21,21,21,22,22,23,23,24,24,25,25,25,26,26,27,27,27,28,28,29,29,

%U 30,30,30,31,31,32,32,32,33,33,33,33

%N Positive integer a is repeated m times, where m is the number of 1-Pythagorean triples (a,b,c) satisfying a<=b.

%C 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.

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

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

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

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

%C ...

%C 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.

%C   ...

%C Related sequences (k-Pythagorean triples):

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%C -4......(1,4,1).............A196376..A196377..A196378

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

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

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

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

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

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

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

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

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

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

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

%C ...

%C Related sequences (primitive k-Pythagorean triples):

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%C -4......(1,4,1).............A196379..A196380..A196381

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

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

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

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

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

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

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

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

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

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

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

%C From _Georg Fischer_, Oct 26 2020: (Start)

%C 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:

%C         A196112 A196113 A196114 - non-primitive 3-Pythagorean

%C     49:      29    1008    1051

%C         A196241 A196242 A196243 - primitive 10-Pythagorean

%C     31:      13     950    1013

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

%H Robert Israel, <a href="/A195770/b195770.txt">Table of n, a(n) for n = 1..10000</a>

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

%e described above, are as follows:

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

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

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

%e 7,8,13

%e 7,33,37

%e 9,15,21

%e 9,56,61

%e 10,32,38

%p f:= proc(a) local F,r,u,b;

%p     r:= 3*a^2;

%p     nops(select(proc(t) local b; b:= (r/t - t - 2*a)/4;

%p (t + r/t) mod 4 = 0 and b::integer and b >= a end proc, numtheory:-divisors(3*a^2)));

%p end proc:

%p seq(a$f(a),a=1..100); # _Robert Israel_, Jul 04 2024

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

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

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

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

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

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

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

%t Flatten[Position[t, {}]]

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

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

%t Table[x[n], {n, 1, z7}]  (* this sequence *)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%K nonn

%O 1,1

%A _Clark Kimberling_, Sep 25 2011

%E Name corrected by _Robert Israel_, Jul 04 2024