A084647 -id:A084647 - cp's OEIS frontend

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

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.

A290501 Hypotenuses for which there exist exactly 11 distinct integer triangles.

Original entry on oeis.org

48828125, 97656250, 146484375, 195312500, 292968750, 341796875, 390625000, 439453125, 537109375, 585937500, 683593750, 781250000, 878906250, 927734375, 1025390625, 1074218750, 1123046875, 1171875000, 1318359375, 1367187500, 1513671875, 1562500000, 1611328125

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 11 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eleven.

			a(1) = 48828125 = 5^11, a(5) = 292968750 = 2*3*5^11, a(101) = 9375000000 = 2^6*3*5^11.

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), 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)^11 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.

A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

Original entry on oeis.org

30517578125, 61035156250, 91552734375, 122070312500, 183105468750, 213623046875, 244140625000, 274658203125, 335693359375, 366210937500, 427246093750, 488281250000, 549316406250, 579833984375, 640869140625, 671386718750, 701904296875, 732421875000

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 15 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fifteen.

			a(1) = 30517578125 = 5^15, a(5) = 183105468750 = 2*3*5^15, a(101) = 5859375000000 = 2^6*3*5^15.

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), A290502 (14), 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)^15 for k, p > 0 ordered by increasing values.

A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

Original entry on oeis.org

3814697265625, 7629394531250, 11444091796875, 15258789062500, 22888183593750, 26702880859375, 30517578125000, 34332275390625, 41961669921875, 45776367187500, 53405761718750, 61035156250000, 68664550781250, 72479248046875, 80108642578125, 83923339843750

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 18 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eighteen.

			a(1) = 3814697265625 = 5^18, a(5) = 22888183593750 = 2*3*5^18, a(101) = 732421875000000 = 2^6*3*5^18.

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), A290502 (14), A290503 (15), A097238 (16), A097239 (17), 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)^18 for k, p > 0 ordered by increasing values.

A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

Original entry on oeis.org

203125, 265625, 406250, 453125, 531250, 578125, 609375, 640625, 796875, 812500, 828125, 906250, 953125, 1062500, 1140625, 1156250, 1218750, 1281250, 1359375, 1390625, 1421875, 1515625, 1578125, 1593750, 1625000, 1656250, 1703125, 1734375, 1765625, 1812500

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 19 different ways into the sum of two nonzero squares: these are those with exactly two distinct prime divisors of the form 4k+1 with one, and six respective multiplicities, or with only one prime divisor of this form with multiplicity nineteen.

			a(1) = 203125 = 5^6*13, a(5) = 531250 = 2*5^6*17, a(281) = 12796875 = 3^2*5^6*7*13.

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), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), 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 products A004144(k)*A002144(p1)*A002144(p2)^6, or A004144(k)*A002144(p1)^19 for k, p1, p2 > 0 ordered by increasing values.

A025303 Numbers that are the sum of 2 distinct nonzero squares in exactly 2 ways.

Original entry on oeis.org

65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 340, 365, 370, 377, 410, 442, 445, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 580, 585, 610, 625, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 820, 865, 884, 890, 901, 905, 949

Offset: 1

David W. Wilson

nonn

Numbers 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 3, and with no prime divisor of the form 4k+3 to an odd multiplicity. - Jean-Christophe Hervé, Dec 01 2013

			65 = 5*13 = 64+1 = 49 + 16; 85 = 5*17 = 81+4 = 49+16; 125 = 5^3 = 121+4 = 100+25; 130 = 2*5*13 = 121+9 = 81+49. - _Jean-Christophe Hervé_, Dec 01 2013

Donovan Johnson, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares

Cf. A001481, A004431, A004018, A230779 (one way).

Cf. analogs for square decompositions: A084645, A084646, A084647, A084648, A084649.

nn = 949; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i - 1}]; Flatten[Position[t, 2]] (* T. D. Noe, Apr 07 2011 *)

A004018(a(n)) = 16. - Jean-Christophe Hervé, Dec 01 2013

A164282 Hypotenuses of more than two Pythagorean triangles.

Original entry on oeis.org

65, 85, 125, 130, 145, 170, 185, 195, 205, 221, 250, 255, 260, 265, 290, 305, 325, 340, 365, 370, 375, 377, 390, 410, 425, 435, 442, 445, 455, 481, 485, 493, 500, 505, 510, 520, 530, 533, 545, 555, 565, 580, 585, 595, 610, 615, 625, 629, 650, 663, 680, 685, 689

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

Also, hypotenuses c of Pythagorean triangles with legs a and b such that a and b are also the hypotenuses of Pythagorean triangles, where the Pythagorean triples (x1,y1,a) and (x2,y2,b) are similar triangles, but the Pythagorean triples (a,b,c) and (x1,y1,a) are not similar. For example, 65^2 = 25^2 + 60^2 with 25^2 = 15^2 + 20^2 and 60^2 = 36^2 + 48^2 with the two smaller triangles being similar. - Naohiro Nomoto

			65 is included because there are 4 distinct Pythagorean triangles with hypotenuse 65. In particular, 65^2 = 16^2 + 63^2 = 25^2 + 60^2 = 33^2 + 56^2 = 39^2 + 52^2.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A009177, A084646, A084647, A084648, A084649.

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,5*5!}];lst

ok(n)={my(t=0); for(k=1, sqrtint(n^2\2), t += issquare(n^2-k^2)); t>2}
select(ok, [1..1000]) \\ Andrew Howroyd, Aug 17 2018

Terms a(45) and beyond from Andrew Howroyd, Aug 17 2018

A232438 Squares or double-squares that are the sum of two distinct nonzero squares in exactly one way.

Original entry on oeis.org

25, 50, 100, 169, 200, 225, 289, 338, 400, 450, 578, 676, 800, 841, 900, 1156, 1225, 1352, 1369, 1521, 1600, 1681, 1682, 1800, 2025, 2312, 2450, 2601, 2704, 2738, 2809, 3025, 3042, 3200, 3362, 3364, 3600, 3721, 4050, 4624, 4900, 5202, 5329, 5408, 5476

Offset: 1

Jean-Christophe Hervé, Dec 01 2013

nonn

Subsequence of A004431 and A001481.

Numbers with exactly one prime factor of form 4k+1 with multiplicity 2, and without prime factor of form 4k+3 to an odd multiplicity.

			25 = 5^2 = 16+9; 50 = 2*5^2 = 49+1.

Jean-Christophe Hervé and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 368 terms from Jean-Christophe Hervé)

Cf. A001481, A004431, A004018, A125853 (A004018 = 4), A230779 (A004018 = 8), A025303 (A004018 = 16).

Analogs for square decompositions: A084645, A084646, A084647, A084648, A084649.

Select[Range[10^4], (IntegerQ[Sqrt[#]] || IntegerQ[Sqrt[#/2]]) && Count[ PowersRepresentations[#, 2, 2], {x_, y_} /; Unequal[0, x, y]] == 1 &]
(* or *) Select[Range[10^4], SquaresR[2, #] == 12 &] (* Jean-François Alcover, Dec 03 2013 *)

A004018(a(n)) = 12.

Terms are obtained by the products A125853(k)*A002144(p)^2 for k, p > 0, ordered by increasing values.

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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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A290501 Hypotenuses for which there exist exactly 11 distinct integer triangles.

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

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A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

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A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

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A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

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A025303 Numbers that are the sum of 2 distinct nonzero squares in exactly 2 ways.

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