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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
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
Sequence in context: A316851 A358470 A196778 * A196008 A004220 A202308
KEYWORD
nonn
AUTHOR
Clark Kimberling, Sep 25 2011
STATUS
approved

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Last modified March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)