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A195770
Positive integer a is repeated m times, where m is the number of 1-Pythagorean triples (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
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
MAPLE
f:= proc(a) local F, r, u, b;
r:= 3*a^2;
nops(select(proc(t) local b; b:= (r/t - t - 2*a)/4;
(t + r/t) mod 4 = 0 and b::integer and b >= a end proc, numtheory:-divisors(3*a^2)));
end proc:
seq(a$f(a), a=1..100); # Robert Israel, Jul 04 2024
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}] (* this sequence *)
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Sep 25 2011
EXTENSIONS
Name corrected by Robert Israel, Jul 04 2024
STATUS
approved