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A181045 Decimal expansion of A060295/24. 3

%I #20 Sep 08 2022 08:45:54

%S 1,0,9,3,9,0,5,8,8,6,0,0,3,2,0,3,0,9,9,9,9,9,9,9,9,9,9,9,9,9,6,8,7,5,

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

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

%N Decimal expansion of A060295/24.

%C This real number is close to the prime number 10939058860032031. Also, the only (single) integer values placed in the denominator that will generate 'near-integers' from this relation are the divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24 (cf. A018253). A total of 64 'near-integers' can be obtained from generating powers (1-8) of A060295 and dividing each by one of the divisors of 24. Example: The last (64th) 'near-integer' is A060295^8 = 2.25698985492608864738884...99926422461218840012234... *10^139 (which is split by ... for brevity), the digits of which close to the decimal point are ...218840.012234... . While this does not quite look like a 'near-integer' this is where the pattern of 0's and 9's in the decimal tail cease in the case. See A166532.

%H G. C. Greubel, <a href="/A181045/b181045.txt">Table of n, a(n) for n = 17..10000</a>

%H Math Overflow, <a href="http://mathoverflow.net/questions/4775/why-are-powers-of-exppisqrt163-almost-integers">Questions</a> [From _Mark A. Thomas_, Oct 02 2010]

%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.

%F Equals exp(Pi * sqrt(163))/24.

%e A060295/24 = 10939058860032030.999999999999968753024883257737036639... This is almost the prime 10939058860032031.

%t E^(Pi Sqrt[163])/24

%t RealDigits[Exp[Pi Sqrt[163]]/24, 10, 100][[1]] (* _G. C. Greubel_, Feb 14 2018 *)

%o (PARI) exp(Pi*sqrt(163))/24 \\ _G. C. Greubel_, Feb 14 2018

%o (Magma) R:= RealField(); Exp(Pi*Sqrt(163))/24;

%Y Cf. A166528, A166529, A166530, A166531.

%K cons,nonn

%O 17,3

%A _Mark A. Thomas_, Sep 30 2010

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