A225119 -id:A225119 - cp's OEIS frontend

A085565 Decimal expansion of lemniscate constant A.

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

Offset: 1

N. J. A. Sloane, Jul 06 2003

This number is transcendental by a result of Schneider on elliptic integrals. - Benoit Cloitre, Jan 08 2006
The two lemniscate constants are A = Integral_{x = 0..1} 1/sqrt(1 - x^4) dx and B = Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx. See A076390. - Peter Bala, Oct 25 2019
Also the ratio of generating curve length to diameter of a "Mylar balloon" (see Paulsen). - Jeremy Tan, May 05 2021

			1.3110287771460599052324197949455597068413774757158115814084108519...

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen (1934).
Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale (1937).

G. C. Greubel, Table of n, a(n) for n = 1..5000
John Maxwell Campbell, WZ proofs for lemniscate-like constant evaluations, Integers 21 (2021), Article A107, 15.
S. Khrushchev, Orthogonal polynomials and continued fractions from Euler’s point of view, Encyclopedia of Mathematics and its Applications 122.
Rensley Meulens, A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation, AIP Advances (2022) Vol. 12, 015308.
W. H. Paulsen, What Is the Shape of a Mylar Balloon?, Amer. Math. Monthly 101 (10), (Dec. 1994), pp. 953-958.
J. Todd, The lemniscate constants, Comm. ACM, 18 (1975), 14-19; 18 (1975), 462.
J. Todd, The lemniscate constants, in Pi: A Source Book, pp. 412-417.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Wikipedia, Mylar balloon.
Index entries for transcendental numbers.

Cf. A076390, A225119, A243340.

C := ComplexField(); [Gamma(1/4)^2/(4*Sqrt(2*Pi(C)))]; // G. C. Greubel, Nov 05 2017

Mathematica

RealDigits[ Gamma[1/4]^2/(4*Sqrt[2*Pi]), 10, 99][[1]] (* or *) RealDigits[ EllipticK[-1], 10, 99][[1]] (* Jean-François Alcover, Mar 07 2013, updated Jul 30 2016 *)

PARI

gamma(1/4)^2/4/sqrt(2*Pi)

PARI

K(x)=Pi/2/agm(1,sqrt(1-x)) K(-1) \\ Charles R Greathouse IV, Aug 02 2018

PARI

ellK(I) \\ Charles R Greathouse IV, Feb 04 2025

Formula

Equals (1/4)*(2*Pi)^(-1/2)*GAMMA(1/4)^2.

Equals Integral_{x>=1}dx/sqrt(4x^3-4x). - Benoit Cloitre, Jan 08 2006

Equals Product_(k>=0, [(4k+3)(4k+4)] / [(4k+5)(4k+2)] ) (Gauss). - Ralf Stephan, Mar 04 2008 [corrected by Vaclav Kotesovec, May 01 2020]

Equals Pi/sqrt(8)/agm(1,sqrt(1/2)).

Equals Pi/sqrt(8)*hypergeom([1/2,1/2],[1],1/2).

Product_{m>=1} ((2*m)/(2*m+1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch

From Peter Bala, Mar 09 2015: (Start)

Equals Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.

Continued fraction representations: 2/(1 + 1*3/(2 + 5*7/(2 + 9*11/(2 + ... )))) due to Euler - see Khrushchev, p. 179.

Also equals 1 + 1/(2 + 2*3/(2 + 4*5/(2 + 6*7/(2 + ... )))). (End)

From Peter Bala, Oct 25 2019: (Start)

Equals 1 + 1/5 + (1*3)/(5*9) + (1*3*5)/(5*9*13) + ... = hypergeom([1/2,1],[5/4],1/2) by Gauss's second summation theorem.

Equivalently, define a sequence of rational numbers r(n) recursively by r(n) = (2*n - 3)/(4*n - 3)*r(n-1) with r(1) = 1. Then the constant equals Sum_{n >= 1} r(n) = 1 + 1/5 + 1/15 + 1/39 + 7/663 + 1/221 + 11/5525 + 11/12325 + 1/2465 + .... The partial sum of the series to 100 terms gives 32 correct decimal digits for the constant.

Equals (1*3)/(1*5) + (1*3*5)/(1*5*9) + (1*3*5*7)/(1*5*9*13) + ... = (3/5) * hypergeom([5/2,1],[9/4],1/2). (End)

Equals (3/2)*A225119. - Peter Bala, Oct 27 2019

Equals Integral_{x=0..Pi/2} 1/sqrt(1 + cos(x)^2) dx = Integral_{x=0..Pi/2} 1/sqrt(1 + sin(x)^2) dx. - Amiram Eldar, Aug 09 2020

From Peter Bala, Mar 24 2024: (Start)

An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 1 for k >= 0.

For example, taking k = 0 and k = 1 yields

A = 2/(1 + (1*3)/(2 + (5*7)/(2 + (9*11)/(2 + (13*15)/(2 + ... + (4*n + 1)*(4*n + 3)/(2 + ... )))))) and

A = (1/4)*(5 + (1*3)/(10 + (5*7)/(10 + (9*11)/(10 + (13*15)/(10 + ... + (4*n + 1)*(4*n + 3)/(10 + ... )))))). (End)

A243340 Decimal expansion of 4L/(3Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 03 2014

			1.11283578889876424837523964373206241199199...

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102.

Eric Weisstein's World of Mathematics, Lemniscate Constant.

Cf. A062539 (L), A076390, A085565, A225119 (L/3).

L = Pi^(3/2)/(Sqrt[2]*Gamma[3/4]^2); RealDigits[4*L/(3*Pi), 10, 103] // First

2*sqrt(2*Pi)/(3*gamma(3/4)^2) \\ Stefano Spezia, Nov 27 2024

Equals 2*sqrt(2*Pi)/(3*Gamma(3/4)^2).
From  Peter Bala, Mar 24 2024: (Start)
An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 3 for k >= 0.
For example, taking k = 0 and k = 1 yields
4*L/(3*Pi) = 1 + 1/(6 + (5*7)/(6 + (9*11)/(6 + (13*15)/(6 + ... + (4*n + 1)*(4*n + 3)/(6 + ... ))))) and
4*L/(3*Pi) = 8/(7 + (1*3)/(14 + (5*7)/(14 + (9*11)/(14 + (13*15)/(14 + ... + (4*n + 1)*(4*n + 3)/(14 + ... )))))).
Equals (2/3) * 1/A076390. (End)

A281792 Primes of the form x^2 + p^4 where x > 0 and p is prime.

Original entry on oeis.org

17, 41, 97, 137, 181, 241, 277, 337, 457, 641, 661, 757, 769, 821, 857, 881, 977, 1109, 1201, 1237, 1301, 1409, 1697, 2017, 2069, 2389, 2417, 2437, 2617, 2657, 2741, 2801, 3041, 3217, 3301, 3329, 3541, 3557, 3697, 3761, 3989, 4001, 4177, 4241, 4337, 4517, 4721, 5557, 5641, 5857, 6101, 6257, 6481, 6577

Offset: 1

Robert Israel, Jan 30 2017

nonn

Heath-Brown and Li prove an asymptotic formula for the number of terms <= x, in particular showing that the sequence is infinite.

			17 = 1^2 + 2^4
41 = 5^2 + 2^4
97 = 9^2 + 2^4
137 = 11^2 + 2^4
181 = 10^2 + 3^4

Robert Israel, Table of n, a(n) for n = 1..10000
D.R. Heath-Brown and X. Li, Prime values of a^2+p^4, arXiv:1504.00531 [math.NT], 2015; Inventiones Mathematicae page 1-59 (Sep 30 2016), doi:10.1007/s00222-016-0694-0

Subsequence of A028916.

Cf. A199401, A225119.

N:= 10000: # to get all terms <= N
A:= select(isprime, {seq(seq(x^2+y^4, x=1..floor(sqrt(N-y^4))),
y=select(isprime, [$1..floor(N^(1/4))]))}):
sort(convert(A,list)); # Robert Israel, Jan 30 2017

nn = 10000;
Select[Table[x^2+y^4, {y, Select[Range[nn^(1/4)], PrimeQ]}, {x, Sqrt[nn-y^4 ]}] // Flatten, PrimeQ] // Union (* Jean-François Alcover, Sep 18 2018, after Robert Israel *)

list(lim)=if(lim<17, return([])); my(v=List(),p4,t); forstep(a=1,sqrtint(-16+lim\=1),2, if(isprime(t=a^2+16), listput(v,t))); forprime(p=3,sqrtnint(lim-4,4), p4=p^4; forstep(a=2,sqrtint(lim-p4),2, if(isprime(t=p4+a^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 13 2017

Heath-Brown and Li prove that there are c*x^(3/4)/log^2 x terms up to x, where c = 4*nu*J = 4.79946121442200811438003177..., nu = A199401, and J = A225119. - Charles R Greathouse IV, Aug 21 2017

A371861 Decimal expansion of Integral_{x=0..1} sqrt(1 - x^3) dx.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 09 2024

			0.8413092631952725567050114474301764812778...

Decimal expansions of Integral_{x=0..1} sqrt(1 - x^k) dx: A003881 (k=2), this sequence (k=3), A225119 (k=4).

Cf. A002161, A073005, A118292, A203131.

RealDigits[Sqrt[Pi] Gamma[1/3]/(6 Gamma[11/6]), 10, 103][[1]]
RealDigits[Sqrt[3] * Gamma[1/3]^3 / (5*Pi*2^(4/3)), 10, 103][[1]] (* Vaclav Kotesovec, Apr 09 2024 *)

intnum(x=0, 1, sqrt(1 - x^3)) \\ Michel Marcus, Apr 10 2024

Equals sqrt(Pi) * Gamma(1/3) / (6 * Gamma(11/6)).
Equals sqrt(3) * Gamma(1/3)^3 / (5*Pi*2^(4/3)). - Vaclav Kotesovec, Apr 09 2024
Equals 3*A118292/10. - Hugo Pfoertner, Apr 09 2024

cp's OEIS Frontend

A085565 Decimal expansion of lemniscate constant A.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A243340 Decimal expansion of 4*L/(3*Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A281792 Primes of the form x^2 + p^4 where x > 0 and p is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A371861 Decimal expansion of Integral_{x=0..1} sqrt(1 - x^3) dx.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A243340 Decimal expansion of 4L/(3Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.