A046080 -id:A046080 - 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.

A024362 Number of primitive Pythagorean triangles with hypotenuse n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

David W. Wilson

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times C takes value n.
a(A137409(n)) = 0; a(A008846(n)) > 0; a(A120960(n)) = 1; a(A024409(n)) > 1; a(A159781(n)) = 4. - Reinhard Zumkeller, Dec 02 2012
If the formula given below is used one is sure to find all a(n) values for hypotenuses n <= N if the summation indices r and s are cut off at rmax(N) = floor((sqrt(N-4)+1)/2) and smax(N) = floor(sqrt(N-1)/2). a(n) is the number of primitive Pythagorean triples with hypotenuse n modulo catheti exchange. - Wolfdieter Lang, Jan 10 2016

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 116-117, 1966.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A020882, A024361, A046079, A046080.
Cf. A000290, A010052.
Cf. A001221, A002144, A106594.

a024362 n = sum [a010052 y | x <- takeWhile (< nn) $ tail a000290_list,
                             let y = nn - x, y <= x, gcd x y == 1]
            where nn = n ^ 2
-- Reinhard Zumkeller, Dec 02 2012

f:= proc(n) local F;
   F:= numtheory:-factorset(n);
   if map(t -> t mod 4, F) <> {1} then return 0 fi;
   2^(nops(F)-1)
end proc:
seq(f(n),n=1..100); # Robert Israel, Jan 11 2016

Table[a0=IntegerExponent[n,2]; If[n==1 || a0>0, cnt=0, m=n/2^a0; p=Transpose[FactorInteger[m]][[1]]; c=Count[p, _?(Mod[#,4]==1 &)]; If[c==Length[p], cnt=2^(c-1), 0]]; cnt, {n,100}]
a[n_] := If[n==1||EvenQ[n]||Length[Select[FactorInteger[n], Mod[#[[1]], 4]==3 &]] >0, 0, 2^(Length[FactorInteger[n]]-1)]; Array[a, 100] (* Frank M Jackson, Jan 28 2018 *)

a(n)={my(m=0,k=n,n2=n*n,k2,l2);
while(1,k=k-1;k2=k*k;l2=n2-k2;if(l2>k2,break);if(issquare(l2),if(gcd(n,k)==1,m++)));  return(m);} \\ Stanislav Sykora, Mar 23 2015

a(n) = [q^n] T(q), n >= 1, where T(q) = Sum_{r>=1,s>=1} rpr(2*r-1, 2*s)*q^c(r,s), with rpr(k,l) = 1 if gcd(k,l) = 1, otherwise 0, and c(r,s) = (2*r-1)^2 + (2s)^2. - Wolfdieter Lang, Jan 10 2016
If all prime factors of n are in A002144 then a(n) = 2^(A001221(n)-1), otherwise a(n) = 0. - Robert Israel, Jan 11 2016
a(4*n+1) = A106594(n), other terms are 0. - Andrey Zabolotskiy, Jan 21 2022

A055527 Shortest other leg of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

4, 3, 12, 8, 24, 6, 12, 24, 60, 5, 84, 48, 8, 12, 144, 24, 180, 15, 20, 120, 264, 7, 60, 168, 36, 21, 420, 16, 480, 24, 44, 288, 12, 15, 684, 360, 52, 9, 840, 40, 924, 33, 24, 528, 1104, 14, 168, 120, 68, 39, 1404, 72, 48, 33, 76, 840, 1740, 11, 1860, 960, 16, 48, 72

Offset: 3

Henry Bottomley, May 22 2000

nonn

From Alex Ratushnyak, Mar 30 2014: (Start)
Least positive k such that n^2 + k^2 is a square.
For odd n, a(n) <= 4*triangular((n-1)/2), because n^2 + (4 * triangular((n-1)/2))^2 = ((n^2+1)/2) ^ 2, which is a perfect square since n is odd.
For n = 4*k+2, a(n) <= 8*triangular(k), because (4k+2)^2 + (4*k*(k+1))^2 = (4*k^2 + 4*k + 2)^2. (End)

T. D. Noe, Table of n, a(n) for n = 3..1000

Cf. A000290, A000217, A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055525, A055526.

See A082183 for a similar sequence involving triangular numbers.

Table[k = 1; While[! IntegerQ[Sqrt[n^2 + k^2]], k++]; k, {n, 3, 100}] (* T. D. Noe, Apr 02 2014 *)

a(n) = sqrt(A055526(n)^2-n^2) = 2*A054436/n.

A055523 Longest other leg of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

4, 3, 12, 8, 24, 15, 40, 24, 60, 35, 84, 48, 112, 63, 144, 80, 180, 99, 220, 120, 264, 143, 312, 168, 364, 195, 420, 224, 480, 255, 544, 288, 612, 323, 684, 360, 760, 399, 840, 440, 924, 483, 1012, 528, 1104, 575, 1200, 624, 1300, 675, 1404, 728, 1512, 783

Offset: 3

Henry Bottomley, May 22 2000

Todd Silvestri, Table of n, a(n) for n = 3..1002
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055524, A055525, A055526, A055527.

seq(`if`(n::even, (n/2-1)*(n/2+1), (n-1)*(n+1)/2), n=3..100); # Robert Israel, Dec 16 2014

a[n_Integer/;n>=3]:=(3 (n^2-2)+(-1)^(n+1) (n^2+2))/8 (* Todd Silvestri, Dec 16 2014 *)

Vec(x^3*(x^3-3*x-4)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 15 2014

a(n) = 2*A055522(n)/n = sqrt(A055524(n)^2-n^2).
a(2k) = (k-1)*(k+1), a(2k+1) = 2k*(k+1).
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: x^3*(x^3-3*x-4) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 15 2014
a(n) = (3*(n^2-2)+(-1)^(n+1)*(n^2+2))/8. - Todd Silvestri, Dec 16 2014
E.g.f.: 1 + (3*x^2/8 + 3*x/8 - 3/4)*exp(x) + (-x^2/8 + x/8 - 1/4)*exp(-x). - Robert Israel, Dec 16 2014

cp's OEIS Frontend

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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A024362 Number of primitive Pythagorean triangles with hypotenuse n.

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