The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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