A084648 -id:A084648 - cp's OEIS frontend

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.

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 *)

A097238 Numbers j that are the hypotenuse of exactly 16 distinct integer-sided right triangles, i.e., j^2 can be written as a sum of two squares in 16 ways.

Original entry on oeis.org

40625, 53125, 81250, 90625, 106250, 115625, 121875, 128125, 159375, 162500, 165625, 181250, 190625, 212500, 228125, 231250, 243750, 256250, 271875, 278125, 284375, 303125, 315625, 318750, 325000, 331250, 340625, 346875, 353125, 362500

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), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

More terms from Ray Chandler, Sep 18 2004

Offset corrected by Michel Marcus, Aug 04 2017

A097239 Numbers n that are the hypotenuse of exactly 17 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 17 ways.

Original entry on oeis.org

21125, 36125, 42250, 54925, 63375, 72250, 84500, 105125, 108375, 109850, 122825, 126750, 144500, 147875, 164775, 169000, 171125, 190125, 210125, 210250, 216750, 219700, 232375, 245650, 252875, 253500, 289000, 295750, 315375, 325125, 329550

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), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

More terms from Ray Chandler, Sep 18 2004

Offset corrected by Michel Marcus, Aug 04 2017

A097244 Numbers n that are the hypotenuse of exactly 31 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 31 ways.

Original entry on oeis.org

27625, 47125, 55250, 60125, 61625, 66625, 78625, 82875, 86125, 87125, 94250, 99125, 110500, 112625, 118625, 120250, 123250, 129625, 133250, 134125, 141375, 144625, 148625, 155125, 157250, 157625, 164125, 165750, 172250, 174250, 177125

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), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097245 (37), A097282 (40), A097626 (67).

A097245 Numbers k that are the hypotenuse of exactly 37 distinct integer-sided right triangles, i.e., k^2 can be written as a sum of two squares in 37 ways.

Original entry on oeis.org

71825, 93925, 122525, 143650, 156325, 173225, 187850, 209525, 215475, 223925, 244205, 245050, 257725, 267325, 273325, 281775, 287300, 296225, 308425, 312650, 346450, 357425, 367575, 375700, 376025, 382925, 409825, 419050, 426725, 430950

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), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097282 (40), A097626 (67).

More terms from Ray Chandler, Sep 18 2004

A097282 Numbers k that are the hypotenuse of exactly 40 distinct integer-sided right triangles, i.e., k^2 can be written as a sum of two squares in 40 ways.

Original entry on oeis.org

32045, 40885, 45305, 58565, 64090, 67405, 69745, 77285, 80665, 81770, 90610, 91205, 96135, 98345, 98605, 99905, 101065, 107185, 111605, 114985, 117130, 120445, 122655, 124865, 127465, 128180, 128945, 130645, 134810, 135915, 137605

Offset: 1

James R. Buddenhagen, Sep 17 2004

nonn

k^2 is always the sum of k^2 and 0^2, but no real triangle can have a zero-length side. Thus, the Mathematica program below searches for length 41 and implicitly drops the zero-squared-plus-n-squared solution. - Harvey P. Dale, Dec 09 2010

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4j+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), 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), A097626 (67).

Select[Range[150000],Length[PowersRepresentations[#^2,2,2]]==41&] (* Harvey P. Dale, Dec 09 2010 *)

A097626 Numbers k that are the hypotenuse of exactly 67 distinct integer-sided right triangles, i.e., k^2 can be written as a sum of two squares in 67 ways.

Original entry on oeis.org

160225, 204425, 226525, 292825, 320450, 337025, 348725, 386425, 403325, 408850, 416585, 453050, 456025, 480675, 491725, 493025, 499525, 505325, 531505, 535925, 544765, 558025, 574925, 585650, 588965, 602225, 613275, 624325, 637325, 640900

Offset: 1

James R. Buddenhagen, Sep 20 2004

nonn

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4j+3. - Chai Wah Wu, Feb 29 2016

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Chai Wah Wu)

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), A097103 (22), A097244 (31), A097245 (37), A097282 (40).

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

More terms from Ray Chandler, Sep 21 2004

cp's OEIS Frontend

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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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.

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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.

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A097238 Numbers j that are the hypotenuse of exactly 16 distinct integer-sided right triangles, i.e., j^2 can be written as a sum of two squares in 16 ways.

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A097239 Numbers n that are the hypotenuse of exactly 17 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 17 ways.

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A097244 Numbers n that are the hypotenuse of exactly 31 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 31 ways.

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A097245 Numbers k that are the hypotenuse of exactly 37 distinct integer-sided right triangles, i.e., k^2 can be written as a sum of two squares in 37 ways.

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Extensions

A097282 Numbers k that are the hypotenuse of exactly 40 distinct integer-sided right triangles, i.e., k^2 can be written as a sum of two squares in 40 ways.

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