A161771 Decimal expansion of (70*exp(Pi*sqrt(163)))^2.

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

Offset: 39

Author

Mark A. Thomas, Jun 18 2009

Keywords

• nonn
• cons

Comments

Where exp^(Pi*sqrt163) is the Ramanujan constant and 70^2 is related to the norm vector 0 of the Leech lattice where 1^2 + 2^2 + 3^2 + ... + 22^2 + 23^2 + 24^2 = 70^2. A curiosity is: exp^2(Pi*sqrt163)*70^2 ~ hc/piGm^2 where all physics values are CODATA 2006 and m = neutron mass and exp^2(Pi*sqrt163)*70^2 = 3.377368...x 10^38 and hc/piGm^2 = 3.37700 x 10^38 (+- 0.00050) where 0.00050 = u_c which is the combined standard uncertainty.

This can also be expressed in a symmetric form in terms of the square of the neutron mass in units of Planck mass: where hc/2PiGm^2 = (Mp/m)^2 (Mp = Planck mass and m = neutron mass) and (exp^2(Pi*sqrt163)70^2)/2 ~ (Mp/m)^2. Note the divisor 2 in this case, which yields (exp^2(Pi*sqrt163)*70^2)/2 = 168868437938467735733159816253012231600.00000040115967. - Mark A. Thomas, Jul 02 2009

Examples

			337736875876935471466319632506024463200.00000080231935662524957710...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 39..10038
• R. Munafo, Notable Properties of Specific Numbers
• M. A. Thomas, Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies, HAL preprint Id: hal-01232022, 2015.
• M. A. Thomas, Number Theoretic Structural Approach to Dimensionless Physics Forms, HAL preprint Id: hal-01580821 [math.NT], 2017.

Crossrefs

Near relation to A160514 and A160515.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(2*Pi(R)*Sqrt(163))*70^2; // G. C. Greubel, Oct 24 2018

Maple

evalf((70*exp(Pi*sqrt(163)))^2,120); # Muniru A Asiru, Oct 25 2018

Mathematica

First@ RealDigits[Exp[Pi Sqrt[163]]^2 70^2, 10, 105] (* Mark A. Thomas, Jun 18 2009, edited by Michael De Vlieger, Feb 19 2018 *)

PARI

default(realprecision, 100); exp(2*Pi*sqrt(163))*70^2 \\ G. C. Greubel, Oct 24 2018

Formula

Equals exp(2*Pi*sqrt(163))*70^2.