A009004 -id:A009004 - cp's OEIS frontend

A156681 Consider all Pythagorean triangles A^2 + B^2 = C^2 with A < B < C; sequence gives values of B, sorted to correspond to increasing A (A009004).

4, 12, 8, 24, 15, 12, 40, 24, 60, 16, 35, 84, 48, 20, 36, 112, 30, 63, 144, 24, 80, 180, 21, 48, 99, 28, 72, 220, 120, 264, 32, 45, 70, 143, 60, 312, 168, 36, 120, 364, 45, 96, 195, 420, 40, 72, 224, 480, 60, 126, 255, 44, 56, 180, 544, 288, 84, 120, 612, 48, 77, 105

Offset: 1

Ant King, Feb 17 2009

The ordered sequence of B values is A009012(n) (allowing repetitions) and A009023(n) (excluding repetitions).

			As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=4, a(2)=12, a(3)=8 and a(4)=24.

Albert H. Beiler, Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
W. Sierpinski, Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Shujing Lyu, Table of n, a(n) for n = 1..5000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
Robert Recorde, The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers, London, 1557. See p. 57.

Cf. A156682, A009004, A009012, A009023.

PythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = sqrt(A156682(n)^2 - A009004(n)^2).

A156682 Consider all Pythagorean triangles A^2 + B^2 = C^2 with AA009004(n)).

Original entry on oeis.org

5, 13, 10, 25, 17, 15, 41, 26, 61, 20, 37, 85, 50, 25, 39, 113, 34, 65, 145, 30, 82, 181, 29, 52, 101, 35, 75, 221, 122, 265, 40, 51, 74, 145, 65, 313, 170, 45, 123, 365, 53, 100, 197, 421, 50, 78, 226, 481, 68, 130, 257, 55, 65, 183, 545, 290, 91, 125, 613, 60, 85, 111

Offset: 1

Ant King, Feb 17 2009

The corresponding sequence for primitive triples is A156679. For all triples, the ordered sequence of C values is A020882 (allowing repetitions) and A009003 (excluding repetitions).

			As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=5, a(2)=13, a(3)=10 and a(4)=25.

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.

Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A009004, A156681, A156679, A020882, A009003.

PythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = sqrt(A009004(n)^2 + A156681(n)^2).

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

Clark Kimberling, Sep 25 2011

nonn

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)

			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

Robert Israel, Table of n, a(n) for n = 1..10000

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

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 *)

Name corrected by Robert Israel, Jul 04 2024

A020884 Ordered short legs of primitive Pythagorean triangles.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 20, 21, 23, 24, 25, 27, 28, 28, 29, 31, 32, 33, 33, 35, 36, 36, 37, 39, 39, 40, 41, 43, 44, 44, 45, 47, 48, 48, 49, 51, 51, 52, 52, 53, 55, 56, 57, 57, 59, 60, 60, 60, 61, 63, 64, 65, 65, 67, 68, 68, 69, 69, 71, 72, 73, 75, 75, 76, 76, 77

Offset: 1

Clark Kimberling

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of A, sorted.
Union of A081874 and A081925. - Lekraj Beedassy, Jul 28 2006
Each term in this sequence is given by f(m,n) = m^2 - n^2 or g(m,n) = 2mn where m and n are relatively prime positive integers with m > n, m and n not both odd. For example, a(1) = f(2,1) = 2^2 - 1^2 = 3 and a(4) = g(4,1) = 2*4*1 = 8. - Agola Kisira Odero, Apr 29 2016
All powers of 2 greater than 4 (2^2) are terms, and are generated by the function g(m,n) = 2mn. - Torlach Rush, Nov 08 2019

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
P. Alfeld, Pythagorean Triples (broken link)
Nick Exner, Generating Pythagorean Triples. This was originally a Java applet (1998), modified by Michael McKelvey in 2001 and redone as an HTML page with JavaScript by Evan Ramos in 2014.
W. A. Kehowski, Pythagorean Triples.
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A009004, A020882, A020883, A020885, A020886. Different from A024352.
Cf. A024359 (gives the number of times n occurs).
Cf. A037213.

a020884 n = a020884_list !! (n-1)
a020884_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 : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

shortLegs = {}; amx = 99; Do[For[b = a + 1, b < (a^2/2), c = (a^2 + b^2)^(1/2); If[c == IntegerPart[c] && GCD[a, b, c] == 1, AppendTo[shortLegs, a]]; b = b + 2], {a, 3, amx}]; shortLegs (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Take[Union[Sort/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@(Select[Subsets[Range[1,101,2],{2}],GCD@@#==1&]))][[;;,1]],80] (* Harvey P. Dale, Feb 06 2025 *)

Extended and corrected by David W. Wilson

A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.

Original entry on oeis.org

60, 480, 780, 1620, 2040, 3840, 4200, 6240, 7500, 12180, 12960, 14760, 15540, 16320, 20580, 21060, 30720, 33600, 40260, 43740, 49920, 55080, 60000, 65520, 66780, 79860, 92820, 97440, 97500, 103680, 113400, 118080, 120120, 124320, 130560, 131820, 164640

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

It is an open question whether any two distinct Pythagorean Triples can have the same product of their sides.
From Amiram Eldar, Nov 22 2020: (Start)
Named after the French writer Antoine de Saint-Exupéry (1900-1944).
The problem of finding two distinct Pythagorean triples with the same product was proposed by Eckert (1984). It is equivalent of finding a nontrivial solution of the Diophantine equation x*y*(x^4-y^4) = z*w*(z^4-w^4) (Bremner and Guy, 1988). (End)

			a(1) = 3*4*5 = 60.

Richard K. Guy, "Triangles with Integer Sides, Medians and Area." D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.
Antoine de Saint-Exupéry, Problème du Pharaon, Liège : Editions Dynamo, 1957.

T. D. Noe, Table of n, a(n) for n = 1..10000
Andrew Bremner and Richard K. Guy, A Dozen Difficult Diophantine Dilemmas, The American Mathematical Monthly, Vol. 95, No. 1 (1988), pp. 31-36.
Ernest J. Eckert, Problem 994, Crux Mathematicorum, Vol. 10, No. 10 (1984), p. 318, entire issue.
Richard K. Guy, Comment to Problem 994, Crux Mathematicorum, Vol. 12, No. 5 (1986), p. 109, entire issue.
Henry Plane, Calcule-moi un parallélépipède..., AMPEP, PLOT No. 22 (2002), pp. 22-23.
Giovanni Resta, Saint-Exupery numbers.
Antoine de Saint Exupéry, Le Problème du Pharaon, Succession Saint Exupéry - d'Agay, 2018.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A009004, A009012, A009111, A020886, A057097, A057098, A057099, A057100.

k=5000000; lst={}; Do[Do[If[IntegerQ[a=Sqrt[c^2-b^2]], If[a>=b, Break[]]; x=a*b*c; If[x<=k, AppendTo[lst,x]]], {b,c-1,4,-1}], {c,5,400,1}]; Union@lst (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009 *)

a(n) = 60*A057097(n) = A057098(n)*A057099(n)*A057100(n).

A195778 Positive integers a for which there is a (-1)-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25

Offset: 1

Clark Kimberling, Sep 25 2011

nonn

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

			First five (-1)-Pythagorean triples (A195778):
(1,1,1), (2,2,2), (3,3,3), (3,8,7), (4,4,4).
First five primitive (-1)-Pythagorean triples (A195796):
(1,1,1), (3,8,7), (5,8,7), (5,21,19), (7,15,13).

Cf. A195770, A009004, A195794, A195795, A195796, A195862, A195863.

z8 = 800; 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}]  (* A195778 *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}]  (* A195794 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}]  (* A195795 *)
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]] (* A195796 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]] (* A195862 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]] (* A195863 *)

A210503 Numbers k that form a primitive Pythagorean triple with k' and sqrt(k^2 + k'^2), where k' is the arithmetic derivative of k.

Original entry on oeis.org

15, 35, 143, 323, 899, 1763, 3599, 4641, 5183, 10403, 11663, 13585, 19043, 22499, 32399, 35581, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 446641, 622081, 656099, 675683

Offset: 1

Paolo P. Lava, Jan 25 2013

nonn

A037074 is a subsequence of this sequence.
If k is the product of a pair of twin primes we have k=p(p+2), k'=2(p+1) and sqrt(k^2+k'^2)=(p+1)^2+1=p(p+2)+2=k+2. These numbers are relatively prime and therefore they form a primitive Pythagorean triple.
Also in the sequence are the following numbers with four distinct prime factors:
        4641 = 3*7*13*17       [form p(p+4)*q(q+4)],
       13585 = 5*11*13*19      [form p(p+6)*q(q+6)],
       35581 = 7*13*17*23      [form p(p+6)*q(q+6)],
      446641 = 13*17*43*47     [form p(p+4)*q(q+4)],
      622081 = 17*23*37*43     [form p(p+6)*q(q+6)],
      700321 = 19*29*31*41     [form p(p+10)*q(q+10)],
From Ray Chandler, Jan 25 2017: (Start)
    24887581 = 47*53*97*103    [form p(p+6)*q(q+6)],
    43518577 = 59*67*101*109   [form p(p+8)*q(q+8)],
   115539901 = 83*97*113*127   [form p(p+14)*q(q+14)],
   158682817 = 89*101*127*139  [form p(p+12)*q(q+12)],
   305162941 = 103*113*157*167 [form p(p+10)*q(q+10)],
  1093514641 = 103*107*313*317 [form p(p+4)*q(q+4)],
  1415940061 = 167*193*197*223 [form p(p+26)*q(q+26)].
And one term with six distinct prime factors:
   650344079 = 7*11*37*53*59*73. (End)

			m=57599, m'=480, sqrt(57599^2 + 480^2) = 57601.

Ray Chandler, Table of n, a(n) for n = 1..500 (terms 1..100 from Paolo P. Lava)

Cf. A003415, A009003, A009004, A037074.

with(numtheory);
A210503:= proc(q)
local a,n,p;
for n from 1 to q do
  a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
  if trunc(sqrt(n^2+a^2))=sqrt(n^2+a^2) and gcd(n,gcd(a,n^2+a^2))=1 then print(n); fi;
od; end:
A210503(100000);

from math import sqrt
from sympy import factorint
from gmpy2 import mpz, is_square, gcd
A210503 = []
for n in range(2, 10**5):
    nd = sum([mpz(n*e/p) for p, e in factorint(n).items()])
    if is_square(nd**2+n**2) and gcd(gcd(n, nd), mpz(sqrt(nd**2+n**2))) == 1:
        A210503.append(n) # Chai Wah Wu, Aug 21 2014

A009188 Short leg of more than one Pythagorean triangle.

Original entry on oeis.org

9, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116

Offset: 1

David W. Wilson

nonn

Values of n for which composite n X n magic squares are possible. - J. Lowell, May 20 2010
If n is in the sequence, k*n is in the sequence for all k > 1. So odd semiprimes (A046315) and numbers of the form 4*p where p is an odd prime are core subsequences which give the initial terms of arithmetic progressions in this sequence. - Altug Alkan, Nov 29 2015
Numbers appearing more than once in A009004. - Sean A. Irvine, Apr 20 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001749, A009004, A020884, A046315, A264828.

filter:= proc(n) not isprime(n) and (n::odd or not isprime(n/2)) end proc:
select(filter, [$9 .. 10000]); # Robert Israel, Nov 30 2015

filterQ[n_] := !PrimeQ[n] && (OddQ[n] || !PrimeQ[n/2]);
Select[Range[9, 120], filterQ] (* Jean-François Alcover, Feb 28 2019, from Maple *)

forcomposite(n=9, 1e3, if(n % 2 == 1 || !isprime(n/2), print1(n, ", "))) \\ Altug Alkan, Dec 01 2015

from sympy import primepi
def A009188(n):
    def f(x): return int(n+2+primepi(x)+primepi(x>>1))
    m, k = n+2, f(n+2)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 17 2024

a(n) = A264828(n+2). - Chai Wah Wu, Oct 17 2024

A057098 Shortest side of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 7, 10, 15, 20, 18, 9, 12, 16, 21, 15, 24, 14, 11, 27, 20, 24, 30, 16, 28, 33, 13, 40, 25, 36, 21, 18, 33, 24, 32, 39, 42, 30, 15, 48, 20, 45, 36, 48, 40, 35, 28, 39, 51, 22, 60, 54, 17, 27, 40, 57, 36, 48, 65, 60, 24, 32, 35, 56, 63, 45, 60, 19, 66, 44, 56

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

			a(1)=3 since 3*4*5=60 is smallest possible positive product

Cf. A009004, A009005, A046083, A057096.

maxShortLeg = 66; terms = 71;
r[a_] := {a, b, c} /. {ToRules[Reduce[a <= b < c && a^2+b^2 == c^2, {b, c}, Integers]]};
abc = r /@ Complement[Range[maxShortLeg], {1, 2, 4}] // Flatten[#, 1]&;
SortBy[abc, Times @@ # &][[;; terms, 1]] (* Jean-François Alcover, Nov 21 2019 *)

a(n) =A057096(n)/(A057099(n)*A057100(n)) =sqrt(A057100(n)^2-A057099(n)^2)

A230543 Numbers n that form a Pythagorean quadruple with n', n'' and sqrt(n^2 + n'^2 + n''^2), where n' and n'' are the first and the second arithmetic derivative of n.

Original entry on oeis.org

512, 1203, 3456, 6336, 23328, 42768, 157464, 249753, 288684, 400000, 722718, 1062882, 1948617, 2700000, 4950000, 18225000, 33412500, 105413504, 123018750, 225534375, 312500000, 408918816

Offset: 1

Paolo P. Lava, Oct 25 2013

Tested up to n = 4.09*10^8.

			If n = 6336 then n' = 23808, n'' = 103936 and sqrt(n^2 + n'^2 + n''^2) = 106816.

Eric Weisstein's World of Mathematics, Pythagorean Quadruple

Cf. A003415, A009003, A009004, A068346, A210503, A211176.

Cf. A096907-A096909 and A097263-A097266 for Pythagorean Quadruples.

with(numtheory): P:= proc(q) local a1, a2, n, p;
for n from 2 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);
if type(sqrt(n^2+a1^2+a2^2),integer) then print(n);
fi; od; end: P(10^10);

a(16)-a(18) from Giovanni Resta, Oct 25 2013
a(19) from Ray Chandler, Dec 22 2016
a(20) from Ray Chandler, Dec 31 2016
a(21) from Ray Chandler, Jan 05 2017
a(22) from Ray Chandler, Jan 09 2017

cp's OEIS Frontend

A156681 Consider all Pythagorean triangles A^2 + B^2 = C^2 with A < B < C; sequence gives values of B, sorted to correspond to increasing A (A009004).

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A156682 Consider all Pythagorean triangles A^2 + B^2 = C^2 with AA009004(n)).

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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.

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A020884 Ordered short legs of primitive Pythagorean triangles.

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A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.

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A195778 Positive integers a for which there is a (-1)-Pythagorean triple (a,b,c) satisfying a<=b.

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A210503 Numbers k that form a primitive Pythagorean triple with k' and sqrt(k^2 + k'^2), where k' is the arithmetic derivative of k.

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