A092040 -id:A092040 - cp's OEIS frontend

A175573 Decimal expansion of Pi^(1/4)/Gamma(3/4).

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

Offset: 1

R. J. Mathar, Jul 15 2010

Entry 34 a of chapter 11 of Ramanujan's second notebook. Entry 34 b is A085565.

			1.0864348112133080145753161...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc., Vol. 15, No. 4 (1983), 273-320.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 27, equation 27:13:18 at page 258.
Wikipedia, Theta function.

Cf. A068465, A085565, A092040, A096427, A175574, A247217, A273081, A273082, A273083, A273084, A319332, A327995.

C := ComplexField(); [(Pi(C))^(1/4)/Gamma(3/4)]; // G. C. Greubel, Nov 05 2017

Maple

Pi^(1/4)/GAMMA(3/4) ; evalf(%) ;

Mathematica

RealDigits[ Pi^(1/4)/Gamma[3/4], 10, 105][[1]] (* Jean-François Alcover, Jul 04 2013 *)

PARI

Pi^(1/4)/gamma(3/4) \\ G. C. Greubel, Nov 05 2017

PARI

2*suminf(k=0,exp(-Pi)^(k^2))-1 \\ Hugo Pfoertner, Sep 17 2018

Formula

Equals A092040 / A068465.

Equals Sum_{n=-oo..oo} exp(-Pi*n^2), or also EllipticTheta(3, 0, exp(-Pi)). - Jean-François Alcover, Jul 04 2013

Equals sqrt(A175574). - Amiram Eldar, Jul 04 2023

Equals Gamma(1/4)/(sqrt(2)*Pi^(3/4)). - Vaclav Kotesovec, Jul 04 2023

Equals Product_{k>=1} tanh((1/2 + i/2)*Pi*k), i=sqrt(-1). - _Antonio Graciá Llorente, Mar 20 2024

Equals Product_{k>=0} (1/2)*(((k+1/2)/(k+1))^(1/2)+((k+1)/(k+1/2))^(1/2)). - Antonio Graciá Llorente, Jul 23 2024

Equals (1/A096427)^2 (see Spanier and Oldham at p. 258). - Stefano Spezia, Dec 31 2024

Equals 2*A319332 = 1/A327995. - Hugo Pfoertner, Dec 31 2024

A367842 Decimal expansion of limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Dec 02 2023

Limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(1/n) = sqrt(2*Pi).

			1.23456019539799897381741853007827189474437277093956302475669920823457...

Cf. A010767, A092040, A074962, A367898, A367899.

RealDigits[(2*Pi)^(1/4)/Glaisher, 10, 120][[1]]
Exp[Integrate[x*Log[Gamma[x]], {x, 0, 1}]]

Equals (2*Pi)^(1/4) / A, where A = A074962 is the Glaisher-Kinkelin constant.
Equals A010767 * A092040 / A074962.
Equals exp(Integral_{x=0..1} x*log(Gamma(x)) dx).

A248557 Decimal expansion of (Pi/2)^(1/4)/Gamma(3/4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 15 2014

			0.913579138156116821407242593401222089701963916393469...

MathOverflow, How to calculate the infinite sum of this double series?, 2014.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

Cf. A010767, A068465, A092040, A175573, A222068, A251992.

RealDigits[(Pi/2)^(1/4)/Gamma[3/4], 10, 103] // First

(Pi/2)^(1/4)/gamma(3/4) \\ Michel Marcus, Dec 15 2014

Also equals theta_2(0,exp(-Pi)), where 'theta' is the elliptic theta function.
Equals A175573 / exp(4*A251992/Pi + Pi/4).
Equals Product_{k>=1} tanh(k*Pi/2). - Amiram Eldar, Jun 12 2021

A380099 a(n) is the n-digit numerator of the fraction h/k with h and k coprime positive integers at which abs((h/k)^4-Pi) is minimal.

Original entry on oeis.org

4, 97, 888, 9551, 13549, 505311, 4601995, 87956765, 298132602

Offset: 1

Stefano Spezia, Jan 12 2025

a(1)^4 = 4^4 = 256 corresponds to the numerator of A210621.

It appears that the number of correct decimal digits of Pi obtained from the fraction a(n)/A380100(n) is A130773(n-1) for n > 1 (see Spezia in Links). - Stefano Spezia, Apr 20 2025

			  n               (h/k)^4    approximated value
  -   -------------------    ------------------
  1               (4/3)^4    3.1604938271604...
  2             (97/73)^4    3.1174212867620...
  3           (888/667)^4    3.1415829223858...
  4         (9551/7174)^4    3.1415927852873...
  5       (13549/10177)^4    3.1415926560044...
  ...

Cf. A000796, A092040, A130773, A210621.

Cf. A355622, A364844, A380100 (denominator).

nmax = 3; a = {}; hmin = kmin = 0; For[n = 1, n <= nmax, n++, minim = Infinity; For[h = 10^(n-1), h <10^n, h++, For[k = 1, k < 10^n/Pi^(1/4), k++, If[(dist = Abs[h^4/k^4-Pi]) < minim && GCD[h,k]==1, minim = dist; hmin=h; kmin = k]]]; AppendTo[a, hmin]]; a

a(6)-a(9) from Kritsada Moomuang, Apr 17 2025

A385568 Decimal expansion of Pi^(3/4).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jul 03 2025

			2.35973049241469688757847446452193444245804053693781...

Index entries for transcendental numbers

Cf. A092040, A175476, A196535.

Mathematica
```
RealDigits[Pi^(3/4), 10, 105][[1]]
```

Equals sqrt(A175476).

Equals A092040^3.

cp's OEIS Frontend

A175573 Decimal expansion of Pi^(1/4)/Gamma(3/4).

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A367842 Decimal expansion of limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2).

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A248557 Decimal expansion of (Pi/2)^(1/4)/Gamma(3/4).

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A380099 a(n) is the n-digit numerator of the fraction h/k with h and k coprime positive integers at which abs((h/k)^4-Pi) is minimal.

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A385568 Decimal expansion of Pi^(3/4).

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