A290501 -id:A290501 - cp's OEIS frontend

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.

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

A290499 Hypotenuses for which there exist exactly 8 distinct integer triangles.

Original entry on oeis.org

390625, 781250, 1171875, 1562500, 2343750, 2734375, 3125000, 3515625, 4296875, 4687500, 5468750, 6250000, 7031250, 7421875, 8203125, 8593750, 8984375, 9375000, 10546875, 10937500, 12109375, 12500000, 12890625, 14062500, 14843750, 16406250, 16796875, 17187500

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 8 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eight.

			a(1) = 390625 = 5^8, a(5) = 2343750 = 2*3*5^8, a(101) = 75000000 = 2^6*3*5^8.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

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

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), 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 *)

Terms are obtained by the product A004144(k)*A002144(p)^8 for k, p > 0 ordered by increasing values.

A290500 Hypotenuses for which there exist exactly 9 distinct integer triangles.

Original entry on oeis.org

1953125, 3906250, 5859375, 7812500, 11718750, 13671875, 15625000, 17578125, 21484375, 23437500, 27343750, 31250000, 35156250, 37109375, 41015625, 42968750, 44921875, 46875000, 52734375, 54687500, 60546875, 62500000, 64453125, 70312500, 74218750, 82031250

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 9 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity nine.

			a(1) = 1953125 = 5^9, a(5) = 11718750 = 2*3*5^9, a(101) = 375000000 = 2^6*3*5^9.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

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

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), 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 *)

Terms are obtained by the product A004144(k)*A002144(p)^9 for k, p > 0 ordered by increasing values.

A290502 Hypotenuses for which there exist exactly 14 distinct integer triangles.

Original entry on oeis.org

6103515625, 12207031250, 18310546875, 24414062500, 36621093750, 42724609375, 48828125000, 54931640625, 67138671875, 73242187500, 85449218750, 97656250000, 109863281250, 115966796875, 128173828125, 134277343750, 140380859375, 146484375000, 164794921875

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 14 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fourteen.

			a(1) = 6103515625 = 5^14, a(5) = 36621093750 = 2*3*5^14, a(101) = 1171875000000 = 2^6*3*5^14.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

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

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), 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 *)

Terms are obtained by the product A004144(k)*A002144(p)^14 for k, p > 0 ordered by increasing values.

cp's OEIS Frontend

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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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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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.

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A290499 Hypotenuses for which there exist exactly 8 distinct integer triangles.

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A290500 Hypotenuses for which there exist exactly 9 distinct integer triangles.

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