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%I #37 Sep 08 2022 08:45:45
%S 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,
%T 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,
%U 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
%N Decimal expansion of (70*exp(Pi*sqrt(163)))^2.
%C 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.
%C 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
%H G. C. Greubel, <a href="/A161771/b161771.txt">Table of n, a(n) for n = 39..10038</a>
%H R. Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Notable Properties of Specific Numbers</a>
%H M. A. Thomas, <a href="https://hal.archives-ouvertes.fr/hal-01232022">Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies</a>, HAL preprint Id: hal-01232022, 2015.
%H M. A. Thomas, <a href="https://hal.archives-ouvertes.fr/hal-01580821">Number Theoretic Structural Approach to Dimensionless Physics Forms</a>, HAL preprint Id: hal-01580821 [math.NT], 2017.
%F Equals exp(2*Pi*sqrt(163))*70^2.
%e 337736875876935471466319632506024463200.00000080231935662524957710...
%p evalf((70*exp(Pi*sqrt(163)))^2,120); # _Muniru A Asiru_, Oct 25 2018
%t 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 *)
%o (PARI) default(realprecision, 100); exp(2*Pi*sqrt(163))*70^2 \\ _G. C. Greubel_, Oct 24 2018
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Exp(2*Pi(R)*Sqrt(163))*70^2; // _G. C. Greubel_, Oct 24 2018
%Y Near relation to A160514 and A160515.
%K nonn,cons
%O 39,1
%A _Mark A. Thomas_, Jun 18 2009
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