A009003 -id:A009003 - cp's OEIS frontend

A120210 Integer squares y from the smallest solutions of y^2 = x(a^N - x)(b^N + x) (elliptic line, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a.

20, 30, 156, 600, 420, 1640, 3660, 520, 2590, 7140, 1224, 10920, 8190, 20880, 32580, 4872, 19998, 5220, 48620, 69960, 3150, 41470, 97656, 132860, 19080, 76830, 176820, 230880, 131070, 12740, 296480, 11100, 375156, 52360, 209950, 468540, 64080

Offset: 1

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

nonn

The case x congruent to 0 mod b or b congruent to 0 mod x is frequent (e.g., A120212). Note that the triples a = 3, b = 4, c = 5 and a = 4, b = 3, c = 5 provide a different result for (x, y).

The natural solution is y = c * b * (c-b) and x = b * (c-b) with c hypotenuse in the triple. - Giorgio Balzarotti, Jul 19 2006

			First primitive Pythagorean triple: 3, 4, 5.
Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).
Smallest integer solution: (x, y) = (4,20).
First element in the sequence: y = 20.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 47.

Cf. A009003, A020884, A120211-A120213.

flag:=1; x:=0; # a, b, c primitive Pythagorean triple
while flag=1 do x:=x+1; y2:=x*(a^2-x)*(x+b^2); if (floor(sqrt(y2)))^2=y2 then print(sqrt(y2)); flag:=0; fi; od;

A163366 a(n) = (-1)^floor((prime(n)+2)/2) mod prime(n).

Original entry on oeis.org

1, 1, 4, 1, 1, 12, 16, 1, 1, 28, 1, 36, 40, 1, 1, 52, 1, 60, 1, 1, 72, 1, 1, 88, 96, 100, 1, 1, 108, 112, 1, 1, 136, 1, 148, 1, 156, 1, 1, 172, 1, 180, 1, 192, 196, 1, 1, 1, 1, 228, 232, 1, 240, 1, 256, 1, 268, 1, 276, 280, 1, 292, 1, 1, 312, 316, 1, 336, 1, 348, 352, 1, 1, 372, 1

Offset: 1

Peter Luschny, Jul 25 2009

nonn

Remove the '1's from the sequence to get A152680.

Product modulo p of the quadratic residues of p, where p = prime(n). [Jonathan Sondow, May 14 2010]

			a(4) = 1 because the quadratic residues of prime(4) = 7 are 1, 2, and 4, and 1*2*4 = 8 == 1 (mod 7). - _Jonathan Sondow_, May 14 2010

Carl-Erik Froeberg, On sums and products of quadratic residues, BIT, Nord. Tidskr. Inf.-behandl. 11 (1971) 389-398. [Jonathan Sondow, May 14 2010]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Rahul Gupta, Algorithmic Number Theory, Section 24.5 [_Jonathan Sondow_, May 14 2010]

Cf. A152680, A005098, A002144, A009003.

Cf. A177860, A177863. - Jonathan Sondow, May 14 2010

seq((-1)^iquo(ithprime(i)+2,2) mod ithprime(i),i=1..113);

Table[Mod[ Apply[Times, Flatten[Position[ Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]]], Prime[n]], {n, 1, 80}] (* Jonathan Sondow, May 14 2010 *)

a(n)*A177863(n) == -1 (mod prime(n)), by Wilson's theorem. - Jonathan Sondow, May 14 2010

a(n) = A177860(n) modulo prime(n). - Jonathan Sondow, May 14 2010

A244659 Decimal expansion of 4*K/Pi, a constant appearing in the asymptotic evaluation of the number of non-hypotenuse numbers not exceeding a given bound, where K is the Landau-Ramanujan constant.

Original entry on oeis.org

9, 7, 3, 0, 3, 9, 7, 7, 6, 7, 7, 1, 7, 8, 1, 9, 9, 4, 2, 5, 4, 4, 9, 1, 2, 8, 1, 1, 7, 3, 6, 4, 6, 8, 1, 1, 0, 7, 6, 3, 4, 3, 9, 6, 3, 4, 7, 9, 0, 8, 2, 4, 2, 7, 3, 7, 6, 3, 0, 9, 0, 2, 1, 6, 3, 2, 5, 9, 7, 1, 0, 1, 8, 6, 4, 1, 5, 1, 6, 3, 4, 2, 9, 5, 2, 0, 4, 0, 4, 2, 0, 7, 6, 2, 1, 3, 8, 7, 4, 2

Offset: 0

Jean-François Alcover, Jul 04 2014

			0.973039776771781994254491281173646811...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.3 Landau-Ramanujan Constant, p. 101.

Daniel Shanks, Non-hypotenuse Numbers, Fib. Quart., 13:4 (1975), pp. 319-321.
Eric Weisstein's MathWorld, Landau-Ramanujan Constant

Cf. A004144, A009003, A064533, A088539.

digits = 100; LandauRamanujan[n_] := With[{K = Ceiling[Log[2, n*Log[3, 10]]]}, N[Product[(((1-2^(-2^k))*4^2^k*Zeta[2^k])/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^2^(-k-1), {k, 1, K}]/Sqrt[2], n]]; K = LandauRamanujan[digits+5]; RealDigits[4*K/Pi, 10, digits] // First (* after Victor Adamchik *)

A293283 Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n.

Original entry on oeis.org

6, 9, 18, 40, 42, 68, 75, 90, 99, 105, 122, 126, 130, 174, 192, 196, 225, 251, 257, 288, 315, 325, 330, 350, 405, 490, 499, 504, 516, 528, 546, 550, 576, 614, 651, 665, 684, 726, 735, 744, 849, 882, 900, 920, 936, 974, 1025, 1032, 1036, 1107, 1140, 1183, 1200

Offset: 1

XU Pingya, Oct 04 2017

nonn

For n > 0, k = (n + 1)(2n + 1)^2 is a term in this sequence, because k^2 = (n * (2n + 1)^2)^2 + (2n + 1)^5. Examples: 18, 75, 196, 405, 726, 1183.
When z^2 = x^2 + y^2 (i.e., z = A009003(n)), (z * y^4)^2 = (x * y^4)^2 + (y^2)^5. Thus z * y^4 is a term in this sequence. For example, 1200. More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 1) * z^(5k - 5) is in this sequence.
When z^2 = x^2 + y^3 (i.e., z = A070745(n)), (z * y)^2 = (x * y)^2 + y^5. Thus z * y is in this sequence. E.g. 6, 18, 40, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 4) * z^(5k - 4) is in this sequence.
When z^2 = x^2 + y^4 (i.e., z = A271576(n)), (z * y^3)^2 = (x * y^3)^2 + (y^2)^5. Thus z * y^3 is also in this sequence. E.g. 40, 405, 1107, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 2) * z^(5k - 4) is in this sequence.

			6^2 = 2^2 + 2^5.
9^2 = 7^2 + 2^5.

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

Cf. A009003, A070067, A070745, A228946, A271576, A274035.

c[n_]: = Count[n^2 - Range[(n^2 - 1)^(1/5)]^5, _?(IntegerQ[Sqrt[#]] &)] > 0;
Select[Range[1200], c]

isok(n) = for (k=1, n-1, if (ispower(n^2-k^2, 5), return (1));); return (0); \\ Michel Marcus, Oct 06 2017

A293694 Numbers z such that x^2 + y^8 = z^2 for positive integers x and y.

Original entry on oeis.org

20, 34, 65, 135, 320, 369, 544, 1040, 1095, 1305, 1350, 1404, 1620, 1625, 1746, 1971, 2056, 2160, 2379, 2754, 3060, 3281, 3996, 4100, 4470, 5120, 5265, 5904, 6625, 7825, 7830, 8194, 8575, 8704, 8796, 10250, 10935, 11125, 11700, 12500, 13154, 14500, 15579

Offset: 1

XU Pingya, Oct 16 2017

nonn

Let i, j and k be nonnegative integers, m > n be positive integers. As ((m^2 - n^2)^(4*i+1) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k))^2 + ((m^2 - n^2)^i * (2*m*n)^(j+1) * (m^2 + n^2)^k)^8 = ((m^2 - n^2)^(4*i) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k+1))^2, so that the number of the form (m^2 - n^2)^(4*i) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k+1) is a term.
When (x, y, z) is a solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(4*i+1) * y^(4*j+2) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j+2) * z^(4*k+1)) is a solution of x^2 + y^8 = z^2.
When (x, y, z) is a solution of x^2 + y^6 = z^2 (i.e., z = A293690(n)), (x^(4*i+1) * y^(4*j+1) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j+1) * z^(4*k+1)) is a solution of x^2 + y^8 = z^2.
When (x, y, z) is a solution of x^2 + y^8 = z^2, (x^(4*i+1) * y^(4*j) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j) * z^(4*k+1)) is also a solution of x^2 + y^8 = z^2.

			12^2 + 2^8 = 20^2, 20 is a term.
63^2 + 2^8 = 65^2, 65 is a term.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..13695

Cf. A009003, A228946, A271576, A293283, A293690, A293692.

z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/8)]^8, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[16000], z]

A309812 Odd integers k such that k^2 is arithmetic mean of two other perfect squares.

Original entry on oeis.org

5, 13, 15, 17, 25, 29, 35, 37, 39, 41, 45, 51, 53, 55, 61, 65, 73, 75, 85, 87, 89, 91, 95, 97, 101, 105, 109, 111, 113, 115, 117, 119, 123, 125, 135, 137, 143, 145, 149, 153, 155, 157, 159, 165, 169, 173, 175, 181, 183, 185, 187, 193, 195, 197, 203, 205, 215, 219

Offset: 1

Mohsin A. Shaikh, Aug 18 2019

nonn

			5 is a term because 5^2 = 25 = (1^2 + 7^2)/2.

Intersection of A005408 and A009003.

Select[Range[1, 300, 2], SquaresR[2, 2 #^2] > 4 &] (* Giovanni Resta, Aug 18 2019 *)

isok(n) = {if (n %2, for (i=1, n, x = 2*n^2-i^2; if ((x!=i^2) && (x>0) && issquare(x), return (i));););} \\ Michel Marcus, Aug 18 2019

More terms from Giovanni Resta, Aug 18 2019

A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 0, 0, 6, 0, 0, 5, 0, 9, 0, 8, 0, 0, 12, 0, 0, 0, 0, 7, 10, 0, 0, 20, 18, 0, 0, 0, 16, 21, 0, 12, 0, 15, 24, 9, 0, 0, 0, 27, 0, 0, 0, 0, 14, 24, 20, 28, 0, 33, 0, 0, 40, 0, 36, 11, 0, 0, 0, 16, 0, 0, 32, 0, 42, 0, 0, 48, 24, 21, 0, 0, 30, 0, 48, 0, 18, 0, 0, 13, 0, 60, 0, 39, 54

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) = 3 as the right triangle with sides (3, 4, 5) has hypotenuse n = 5 smallest side a(5) = 3. This is the smallest side a right triangle with integer sides and hypotenuse 5 can have. - _David A. Corneth_, Apr 10 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108708.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_]:=Block[{k=n-1,m=Sqrt[n/2],a},While[k>m&&!IntegerQ[(a=Sqrt[n^2-k^2])],k--];If[k<=m,0,a]];Table[f[n],{n,90}]

first(n) = {my(lh = List(), res = vector(n, i, oo)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = vecmin([res[i*(u^2+v^2)], i*(u^2 - v^2), i*2*u*v]))))); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

Extended by Ray Chandler, Dec 20 2011

A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 12, 0, 12, 0, 15, 0, 0, 16, 0, 0, 0, 0, 24, 24, 0, 0, 21, 24, 0, 0, 0, 30, 28, 0, 35, 0, 36, 32, 40, 0, 0, 0, 36, 0, 0, 0, 0, 48, 45, 48, 45, 0, 44, 0, 0, 42, 0, 48, 60, 0, 0, 0, 63, 0, 0, 60, 0, 56, 0, 0, 55, 70, 72, 0, 0, 72, 0, 64, 0, 80, 0, 0, 84, 0, 63, 0

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) is 4 as the maximum side (other than the hypotenuse) a right triangle with integer sides and hypotenuse 5 can have.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108707.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_] := Block[{k = n - 1, m = Sqrt[n/2]}, While[k > m && !IntegerQ[Sqrt[n^2 - k^2]], k-- ]; If[k <= m, 0, k]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Jun 21 2005 *)

first(n) = {my(lh = List(), res = vector(n)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = max(res[i*(u^2+v^2)], max(i*(u^2 - v^2), i*2*u*v)); ); ); ); ); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

More terms from Robert G. Wilson v, Jun 21 2005

A242219 Smallest a(n) in Pythagorean triple (a, b, c) such that c(n) - b(n) = n.

Original entry on oeis.org

3, 4, 9, 8, 15, 12, 21, 12, 15, 20, 33, 24, 39, 28, 45, 24, 51, 24, 57, 40, 63, 44, 69, 36, 35, 52, 45, 56, 87, 60, 93, 40, 99, 68, 105, 48, 111, 76, 117, 60, 123, 84, 129, 88, 75, 92, 141, 72, 63, 60, 153, 104, 159, 72, 165, 84, 171, 116, 177, 120, 183, 124, 105, 80, 195, 132, 201, 136, 207, 140, 213, 84, 219, 148, 105, 152, 231, 156

Offset: 1

V.J. Pohjola, May 07 2014

nonn

The local minima a(n) predominantly fluctuate, with an increasing amplitude, between the multiples of the leg lengths of the smallest primitive triple (3,4,5) and of its symmetric counterpart (4,3,5). When n grows, minima appear from higher primitive triples which further increase the amplitude.

We have a^2 = c^2 - b^2 = (c-b)(c+b) = n*(c+b). To find the least such square, use n=core(n)*f^2 with core = A007913, f = A000188(n), and look for the least c+b = c-b+2b = n+2b = core(n)*x^2 or x^2 = (n+2b)/core(n) = f^2 + 2b/core(n). The least such integer x is f+1 if core(n) is even, or else f+2. - M. F. Hasler, May 08 2014

			For n=7, a(7) = sqrt(2*7*h(12)-7^2) = 21;
for n=8, a(8) = sqrt(2*8*h(3)-8^2) = 12;
for n=9, a(9) = sqrt(2*9*h(5)-9^2) = 15;
for n=10, a(10) = sqrt(2*10*h(7)-10^2) = 20.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

(* first do *) hypos = A009003; (* then *)
Table[ First[ Union[ Flatten[ Table[ Select[{Sqrt[2 hypos[[i]]*n - n^2]}, IntegerQ && hypos[[i]] > n], {i, 1, Length[hypos]}]]]], {n, 1, 200}]
(* view table *) ListLinePlot[%]

a(n)={ my( f=core(n,1)); sqrtint(( if( bittest( f[1],0), 4*f[2]+4, 2*f[2]+1)*f[1]+n )*n )} \\ M. F. Hasler, May 08 2014

a(n) = min(sqrt(2n*h(i)-n^2), where h(i)=A009003(i)>n.

a(n) = sqrt(( n + (x^2 - f^2)*core(n))*n ) where f = A000188(n), x = f+1 if core(n) = A007913(n) = n / f^2 is even, x = f+2 if core(n) is odd. - M. F. Hasler, May 08 2014

A244662 Decimal expansion of 'C' (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.

Original entry on oeis.org

7, 0, 4, 7, 5, 3, 4, 5, 1, 7, 0, 5, 9, 4, 7, 8, 8, 4, 1, 2, 2, 5, 5, 8, 1, 9, 7, 5, 9, 1, 8, 9, 8, 8, 1, 8, 5, 2, 1, 5, 9, 9, 7, 6, 4, 5, 4, 9, 2, 3, 5, 8, 3, 1, 6, 1, 7, 4, 4, 5, 4, 8, 8, 3, 4, 1, 3, 6, 2, 8, 4, 6, 3, 9, 0, 3, 1, 8, 8, 4, 4, 4, 6, 0, 6, 3, 6, 4, 9, 2, 5, 3, 5, 2, 2, 3, 0, 2, 6, 4

Offset: 0

Jean-François Alcover, Jul 04 2014

			0.70475345170594788412255819759189881852...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.3 Landau-Ramanujan Constant, p. 101.

Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation, Vol. 18 (1964), pp. 75-86.
Daniel Shanks, Non-hypotenuse Numbers, Fib. Quart., 13:4 (1975), pp. 319-321.
Eric Weisstein's MathWorld, Lemniscate Constant

Cf. A009003, A004144, A062539, A227158, A244659 (first order term).

digits = 100; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; c = A227158 = f[m]; c + 1/2 Log[(Pi/L)^2*Exp[EulerGamma]/2] // RealDigits[#, 10, digits] & // First

C = c + 1/2*log((Pi/L)^2*exp(gamma)/2), where c is A227158 and L the Lemniscate constant A062539.

cp's OEIS Frontend

A120210 Integer squares y from the smallest solutions of y^2 = x*(a^N - x)*(b^N + x) (elliptic line, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a.

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A163366 a(n) = (-1)^floor((prime(n)+2)/2) mod prime(n).

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A244659 Decimal expansion of 4*K/Pi, a constant appearing in the asymptotic evaluation of the number of non-hypotenuse numbers not exceeding a given bound, where K is the Landau-Ramanujan constant.

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A293283 Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n.

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A293694 Numbers z such that x^2 + y^8 = z^2 for positive integers x and y.

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Mathematica

A309812 Odd integers k such that k^2 is arithmetic mean of two other perfect squares.

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Mathematica

PARI

Extensions

A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

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Extensions

A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

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A120210 Integer squares y from the smallest solutions of y^2 = x(a^N - x)(b^N + x) (elliptic line, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a.