A212003 -id:A212003 - cp's OEIS frontend

A212002 Decimal expansion of (2*Pi)^2.

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

Offset: 2

Omar E. Pol, Aug 11 2012

This constant appears in Kepler's 3rd Law, T^2 = (2*Pi)^2/GM*a^3 where a is the semi-major axis of a planet orbiting the Sun, T is its period, and GM is the standard gravitational parameter. - Raphie Frank, Dec 13 2012

García & Marco give a generalized zeta regularization by which this is the value of the product of the primes. - Charles R Greathouse IV, Jun 17 2013

			39.4784176043574344753379639995046045412547976289631...

J.-P. Allouche, The zeta-regularized product of odious numbers, arXiv:1906.10532 [math.NT], 2019.
Hyperphysics, Kepler's Laws
E. Muñoz García and R. Pérez Marco, The product over all primes is 4Pi^2, Communications in Mathematical Physics, Vol. 277, No. 1 (2008), pp. 69-81.
Hengguang Li and Jeffrey S. Ovall, A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential, Discrete and Continuous Dynamical Systems Series B, Volume 20, Number 5, July 2015, pp. 1377-1391. Also doi:10.3934/dcdsb.2015.20.1377
Wikipedia, Pendulum.
Index entries for transcendental numbers

Cf. A000796, A002388, A019692, A212003.

RealDigits[(2*Pi)^2,10,120][[1]] (* Harvey P. Dale, Mar 27 2019 *)

4*Pi^2 \\ Charles R Greathouse IV, Jun 17 2013

Equals Product_{k=1..10, gcd(k,10)==1} Gamma(k/10) = Gamma(1/10)*Gamma(3/10)*Gamma(7/10)*Gamma(9/10). - Amiram Eldar, Jun 12 2021

Equals lim_{n->oo} |B(2*n)/B(2*n+2)|*(2*n+1)*(2*n+2), where B(n) denotes the n-th Bernoulli number. - Peter Luschny, Dec 09 2021

A212004 Decimal expansion of (2*Pi)^4.

Original entry on oeis.org

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

Offset: 4

Omar E. Pol, Aug 11 2012

			1558.545456544038995783...

Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200v3, 1997-1998, p. 11.
Index entries for transcendental numbers

Cf. A000796, A019692, A092425, A212002, A212003, A212005.

RealDigits[(2*Pi)^4, 10, 100][[1]] (* Amiram Eldar, Jun 12 2021 *)

16*Pi^4 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{k=1..15, gcd(k,15)==1} Gamma(k/15). - Amiram Eldar, Jun 12 2021

A212005 Decimal expansion of (2*Pi)^5.

Original entry on oeis.org

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

Offset: 4

Omar E. Pol, Aug 11 2012

			9792.629913129006504407721921389939407369622612098399...

Index entries for transcendental numbers

Cf. A000796, A019692, A092731, A212002, A212003, A212004, A212006.

RealDigits[(2*Pi)^5, 10, 100][[1]] (* Amiram Eldar, Jun 12 2021 *)

32*Pi^5 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{k=1..22, gcd(k,22)==1} Gamma(k/22). - Amiram Eldar, Jun 12 2021

A195823 Decimal expansion of 8Pi5^(1/2).

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Feb 01 2012

Equals 4*A212003/Product(Gamma(i/4), i=1..4). - Bruno Berselli, Jan 17 2013

			56.1985178483258111452509971456391583957320734965376193599177682...

Soon-Yi Kang and Chang Heon Kim, Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups, arXiv:0904.3777v1, Apr 24 2009, p. 5 (Example 1).
Index entries for transcendental numbers

Cf. A000796.

RealDigits[8*Pi*5^(1/2), 10, 87][[1]] (* Bruno Berselli, Feb 04 2012 *)

8*Pi*sqrt(5) \\ Charles R Greathouse IV, Oct 01 2022

More terms from Bruno Berselli, Feb 04 2012

A357319 Decimal expansion of 6PiGamma(2/3)^2/(sqrt(3)*Gamma(1/3)^4).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Sep 23 2022

			0.3874382387848854205695648847540189480496...

Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 21.

Cf. A000796, A002194, A073005, A073006, A212003.

(8*Pi^3)/(sqrt(3)*GAMMA(1/3)^6): evalf(%, 92); # Peter Luschny, Sep 24 2022

First[RealDigits[N[6*Pi*Gamma[2/3]^2/(Sqrt[3]*Gamma[1/3]^4), 90]]]

6*Pi*gamma(2/3)^2/(sqrt(3)*gamma(1/3)^4) \\ Michel Marcus, Sep 24 2022

Equals 6*A000796*A073006^2/(A002194*A073005^4).

Equals (8*Pi^3)/(sqrt(3)*Gamma(1/3)^6) = A212003/(A002194*A073005^6). - Peter Luschny, Sep 24 2022

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A212002 Decimal expansion of (2*Pi)^2.

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A212004 Decimal expansion of (2*Pi)^4.

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A212005 Decimal expansion of (2*Pi)^5.

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A195823 Decimal expansion of 8*Pi*5^(1/2).

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A357319 Decimal expansion of 6*Pi*Gamma(2/3)^2/(sqrt(3)*Gamma(1/3)^4).

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A195823 Decimal expansion of 8Pi5^(1/2).

A357319 Decimal expansion of 6PiGamma(2/3)^2/(sqrt(3)*Gamma(1/3)^4).