A097226 -id:A097226 - cp's OEIS frontend

A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

5, 10, 13, 15, 17, 20, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 51, 52, 53, 55, 58, 60, 61, 68, 70, 73, 74, 78, 80, 82, 87, 89, 90, 91, 95, 97, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 135, 136, 137, 140, 143, 146, 148, 149

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is uniquely decomposable into the sum of two nonzero squares: these are those numbers with exactly one prime divisor of the form 4k+1 with multiplicity one. - Jean-Christophe Hervé, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_] := {b, c} /. {ToRules[ Reduce[0 < b < c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[ Range[150], Length[r[#]] == 1 &] (* Jean-François Alcover, Oct 22 2012 *)

is_a084645(n) = #qfbsolve(Qfb(1,0,1),n^2,3)==3 \\ Hugo Pfoertner, Sep 28 2024

Terms are obtained by the products A004144(k)*A002144(p) for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A046080(a(n)) = 1, A046109(a(n)) = 12. - Jean-Christophe Hervé, Dec 01 2013

A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.

Original entry on oeis.org

125, 250, 375, 500, 750, 875, 1000, 1125, 1375, 1500, 1750, 2000, 2197, 2250, 2375, 2625, 2750, 2875, 3000, 3375, 3500, 3875, 4000, 4125, 4394, 4500, 4750, 4913, 5250, 5375, 5500, 5750, 5875, 6000, 6125, 6591, 6750, 7000, 7125, 7375, 7750

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 3 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity three. - Jean-Christophe Hervé, Nov 11 2013

			a(1) = 125 = 5^3, and 125^2 = 100^2 + 75^2 = 117^2 + 44^2 = 120^2 + 35^2. - _Jean-Christophe Hervé_, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1140 terms from Jean-Christophe Hervé)
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==3,AppendTo[lst,n]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^3 for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.

Original entry on oeis.org

65, 85, 130, 145, 170, 185, 195, 205, 221, 255, 260, 265, 290, 305, 340, 365, 370, 377, 390, 410, 435, 442, 445, 455, 481, 485, 493, 505, 510, 520, 530, 533, 545, 555, 565, 580, 585, 595, 610, 615, 625, 629, 663, 680, 685, 689, 697, 715, 730, 740, 745

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 4 different ways into the sum of two nonzero squares: these are those with exactly 2 distinct prime divisors of the form 4k+1, each with multiplicity one, or with only one prime divisor of this form with multiplicity 4. - Jean-Christophe Hervé, Nov 11 2013

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

			a(1) = 65 = 5*13, and 65^2 = 52^2 + 39^2 = 56^2 + 33^2 = 60^2 + 25^2 = 63^2 + 16^2. - _Jean-Christophe Hervé_, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==4,AppendTo[lst,n]],{n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

A084649 Hypotenuses for which there exist exactly 5 distinct Pythagorean triangles.

Original entry on oeis.org

3125, 6250, 9375, 12500, 18750, 21875, 25000, 28125, 34375, 37500, 43750, 50000, 56250, 59375, 65625, 68750, 71875, 75000, 84375, 87500, 96875, 100000, 103125, 112500, 118750, 131250, 134375, 137500, 143750, 146875, 150000, 153125

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 5 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity 5. - Jean-Christophe Hervé, Nov 12 2013

			a(1) = 5^5, a(5) = 6*5^5, a(65) = 13^5. - _Jean-Christophe Hervé_, Nov 12 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1019 terms from Jean-Christophe Hervé)
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==5,AppendTo[lst,n]],{n,3*6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^5 for k, p > 0 ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A084646 Hypotenuses for which there exist exactly 2 distinct integer triangles.

Original entry on oeis.org

25, 50, 75, 100, 150, 169, 175, 200, 225, 275, 289, 300, 338, 350, 400, 450, 475, 507, 525, 550, 575, 578, 600, 675, 676, 700, 775, 800, 825, 841, 867, 900, 950, 1014, 1050, 1075, 1100, 1150, 1156, 1175, 1183, 1200, 1225, 1350, 1352, 1369, 1400

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 2 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity two. - Jean-Christophe Hervé, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==2,AppendTo[lst,n]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^2 for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A046080(a(n)) = 2, A046109(a(n)) = 20. - Jean-Christophe Hervé, Dec 01 2013

A097102 Numbers n that are the hypotenuse of exactly 13 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 13 ways.

Original entry on oeis.org

1105, 1885, 2210, 2405, 2465, 2665, 3145, 3315, 3445, 3485, 3770, 3965, 4420, 4505, 4745, 4810, 4930, 5185, 5330, 5365, 5655, 5785, 5945, 6205, 6290, 6305, 6409, 6565, 6630, 6890, 6970, 7085, 7215, 7345, 7395, 7540, 7565, 7585, 7685, 7735, 7930, 7995

Offset: 1

James R. Buddenhagen, Sep 15 2004

nonn

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

More terms from Ray Chandler, Sep 16 2004

Definition corrected by Zak Seidov, Feb 26 2008

A097101 Numbers n that are the hypotenuse of exactly 7 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 7 ways.

Original entry on oeis.org

325, 425, 650, 725, 845, 850, 925, 975, 1025, 1275, 1300, 1325, 1445, 1450, 1525, 1690, 1700, 1825, 1850, 1950, 2050, 2175, 2225, 2275, 2425, 2525, 2535, 2550, 2600, 2650, 2725, 2775, 2825, 2873, 2890, 2900, 2925, 2975

Offset: 1

James R. Buddenhagen, Sep 15 2004

nonn

Comment from R. J. Mathar, Feb 26 2008, edited by Zak Seidov May 12 2008: (Start)
There are nonsquares x which can be written as a sum of 2 nonzero squares in exactly 7 different ways and which are by definition not in this sequence.
203125 = (125*sqrt(13))^2 is the first example: 203125 = 625 + 202500 = 10404 + 192721 = 18225 + 184900= 22500 + 180625= 62500 + 140625= 69169 + 133956= 84100 + 119025.
The second and third examples are 265625 = (125*sqrt(17))^2 and 406250=(125*sqrt(26))^2. (End)
If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

			Example supplied by _R. J. Mathar_, Feb 26 2008:
The smallest number that can be written as a sum of two nonzero squares in 7 different ways is 105625 = 325^2:
1296 + 104329 = 105625 = 325^2
6400 + 99225 = 105625 = 325^2
8281 + 97344 = 105625 = 325^2
15625 + 90000 = 105625 = 325^2
27225 + 78400 = 105625 = 325^2
38025 + 67600 = 105625 = 325^2
41616 + 64009 = 105625 = 325^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b,c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Equals {n: A025426(n^2)=7}.

Definition and comments corrected by Zak Seidov, Feb 26 2008, May 12 2008

A097103 Numbers n that are the hypotenuse of exactly 22 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 22 ways.

Original entry on oeis.org

5525, 9425, 11050, 12025, 12325, 13325, 14365, 15725, 16575, 17225, 17425, 18785, 18850, 19825, 22100, 22525, 23725, 24050, 24505, 24650, 25925, 26650, 26825, 28275, 28730, 28925, 29725, 31025, 31265, 31450, 31525, 32825, 33150, 34450

Offset: 1

James R. Buddenhagen, Sep 15 2004

nonn

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b,c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

More terms from Ray Chandler, Sep 16 2004

Definition corrected by Zak Seidov, Feb 26 2008

A097219 Numbers n that are the hypotenuse of exactly 6 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 6 ways.

Original entry on oeis.org

15625, 31250, 46875, 62500, 93750, 109375, 125000, 140625, 171875, 187500, 218750, 250000, 281250, 296875, 328125, 343750, 359375, 375000, 421875, 437500, 484375, 500000, 515625, 562500, 593750, 656250, 671875, 687500, 718750, 734375

Offset: 1

James R. Buddenhagen, Sep 17 2004

nonn

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

More terms from Ray Chandler, Sep 18 2004

A097225 Numbers n that are the hypotenuse of exactly 10 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 10 ways.

Original entry on oeis.org

1625, 2125, 3250, 3625, 4250, 4625, 4875, 5125, 6375, 6500, 6625, 7250, 7625, 8500, 9125, 9250, 9750, 10250, 10875, 10985, 11125, 11375, 12125, 12625, 12750, 13000, 13250, 13625, 13875, 14125, 14500, 14625, 14875, 15250, 15375, 17000, 17125

Offset: 1

James R. Buddenhagen, Sep 17 2004

nonn

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[n_] := Reduce[0 < x <= y && n^2 == x^2 + y^2, {x, y}, Integers]; Reap[For[n = 5, n <= 20000, n++, rn = r[n]; If[rn =!= False, If[Length[r[n]] == 10, Print[n]; Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Nov 15 2016 *)

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A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

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A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.

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A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.

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A084649 Hypotenuses for which there exist exactly 5 distinct Pythagorean triangles.

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A084646 Hypotenuses for which there exist exactly 2 distinct integer triangles.

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A097102 Numbers n that are the hypotenuse of exactly 13 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 13 ways.

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A097101 Numbers n that are the hypotenuse of exactly 7 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 7 ways.

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A097103 Numbers n that are the hypotenuse of exactly 22 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 22 ways.

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