A195770
Positive integer a is repeated m times, where m is the number of 1-Pythagorean triples (a,b,c) satisfying a<=b.
Original entry on oeis.org
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
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
-
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
-
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 *)
A156679
Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).
Original entry on oeis.org
5, 13, 25, 17, 41, 61, 37, 85, 113, 65, 145, 181, 29, 101, 221, 265, 145, 313, 365, 53, 197, 421, 481, 257, 65, 545, 613, 85, 325, 685, 89, 761, 401, 841, 925, 125, 485, 1013, 1105, 73, 577, 1201, 149, 1301, 173, 677, 1405, 1513, 785, 185, 1625, 1741, 109, 229
Offset: 1
As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=5, a(2)=13, a(3)=25 and a(4)=17.
- Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
- Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.
-
a156679 n = a156679_list !! (n-1)
a156679_list = f 1 1 where
f u v | v > uu `div` 2 = f (u + 1) (u + 2)
| gcd u v > 1 || w == 0 = f u (v + 2)
| otherwise = w : f u (v + 2)
where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012
-
PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[iHarvey P. Dale, May 10 2020 *)
A263728
Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.
Original entry on oeis.org
3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, 17, 9, 40, 41, 11, 60, 61, 12, 35, 37, 13, 84, 85, 15, 112, 113, 16, 63, 65, 17, 144, 145, 19, 180, 181, 20, 21, 29, 20, 99, 101, 21, 220, 221, 23, 264, 265, 24, 143, 145, 25, 312, 313, 27, 364, 365, 28, 45, 53
Offset: 1
The first few triples are [3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [13, 84, 85], [15, 112, 113], [16, 63, 65], [17, 144, 145], [19, 180, 181], [20, 21, 29], [20, 99, 101], ... - _N. J. A. Sloane_, Dec 15 2015
- H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, Chapter 5, Section 5.3.
- Colin Barker, Table of n, a(n) for n = 1..10000
- Yoshinosuke Hirakawa and Hideki Matsumura, A unique pair of triangles, Journal of Number Theory, Volume 194, January 2019, Pages 297-302. About the remarkable (135,352,377) Pythagorean triple.
- Eric Weisstein's World of Mathematics, Pythagorean Triple
- Wikipedia, Pythagorean triple
-
a:=[]; b:={}; M:=30;
for u from 2 to M do for v from 1 to u-1 do
if gcd(u,v)=1 and u+v mod 2 = 1 then t1:=u^2-v^2; t2:= 2*u*v; t3:=u^2+v^2;
w:=sort([t1,t2]); a:=[op(a), [op(w),t3]]; b:={ op(b), op(w), t3};
fi:
od: od:
a;
sort(a); # A263728
sort(b); # A016825 and A042965 (Maple code from N. J. A. Sloane, Dec 15 2015)
-
\\ Primitive Pythagorean triples (a,b,c) with a
A277557
The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.
Original entry on oeis.org
8, 18, 19, 32, 33, 34, 50, 52, 53, 72, 73, 74, 75, 76, 98, 99, 100, 101, 102, 103, 128, 131, 133, 134, 162, 163, 164, 165, 166, 167, 168, 169, 200, 201, 202, 203, 204, 205, 206, 207, 208, 242, 244, 247, 248, 250, 251, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 338
Offset: 1
a(5)=33 because the ordered pair (2,5) maps to 33 by Cantor's pairing function (see below) and is the 5th such occurrence. Also x=2, y=5 generates a PPT with sides (21,20,29).
Note: Cantor's pairing function is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2, thus A001477(2,5) = 5 + (2+5)*(2+5+1)/2 = 33. - _Antti Karttunen_, Nov 02 2016
Cf.
A020882 (is obtained when
A048147(a(n)) is sorted into ascending order),
A008846 (same with duplicates removed).
Cf.
A020887,
A020888,
A120427,
A024362,
A024406,
A046079,
A046087,
A070151,
A156678,
A156679,
A156680,
A156683,
A156685,
A222946,
A278147.
-
Cantor[{i_, j_}] := (i+j)(i+j+1)/2+j; getparts[n_] := Reverse@Select[Reverse[IntegerPartitions[n, {2}], 2], GCD@@#==1 &]; pairs=Flatten[Table[getparts[2n+1], {n, 1, 20}], 1]; Table[Cantor[pairs[[n]]], {n, 1, Length[pairs]}]
A156680
Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).
Original entry on oeis.org
1, 7, 17, 7, 31, 49, 23, 71, 97, 47, 127, 161, 1, 79, 199, 241, 119, 287, 337, 17, 167, 391, 449, 223, 23, 511, 577, 41, 287, 647, 41, 721, 359, 799, 881, 73, 439, 967, 1057, 7, 527, 1151, 89, 1249, 113, 623, 1351, 1457, 727, 119, 1567, 1681, 31, 161, 839, 1799, 1921
Offset: 1
As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=4-3=1, a(2)=12-5=7, a(3)=24-7=17 and a(4)=15-8=7.
- Barning, F. J. M.; On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices. (Dutch), Math. Centrum Amsterdam Afd. Zuivere Wisk. ZW-001 (1963).
-
PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i
A156689
Inradii of primitive Pythagorean triples a^2+b^2=c^2, 0A020884).
Original entry on oeis.org
1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 6, 9, 10, 11, 11, 12, 13, 10, 13, 14, 15, 15, 12, 16, 17, 14, 17, 18, 15, 19, 19, 20, 21, 18, 21, 22, 23, 15, 23, 24, 21, 25, 22, 25, 26, 27, 27, 24, 28, 29, 21, 26, 29, 30
Offset: 1
The eighth primitive Pythagorean triple ordered by increasing a is (13,84,85). As this has inradius 1/2 (13+84-85)=6, we have a(8)=6.
- Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.
- D. G. Rogers, Putting Pythagoras in the frame, Mathematics Today, The Institute of Mathematics and its Applications, Vol. 44, No. 3, June 2008, pp. 123-125.
-
a156689 n = a156689_list !! (n-1)
a156689_list = f 1 1 where
f u v | v > uu `div` 2 = f (u + 1) (u + 2)
| gcd u v > 1 || w == 0 = f u (v + 2)
| otherwise = (u + v - w) `div` 2 : f u (v + 2)
where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012
-
PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i
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