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%I #44 Sep 08 2022 08:46:05
%S 2,0,5,9,4,0,7,4,0,5,3,4,2,5,7,6,1,4,4,5,3,9,4,7,5,4,9,9,2,3,3,2,7,8,
%T 6,1,2,9,7,7,2,5,4,7,2,6,3,3,5,3,4,0,2,0,9,2,9,9,7,1,8,7,7,9,8,0,5,4,
%U 4,2,8,1,9,6,8,4,6,1,3,5,3,5,7,4,8,1,8,5,7,4,4,8,3,4,9,7,8,2,8,3,1,5,0,1,5
%N Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].
%C Apart from the first digit, the same as A143280. The sum of the reciprocals of the double factorial numbers, Sum_{n>=1} 1/n!! = Sum_{n>=2} n!!/n!. - _Robert G. Wilson v_, Jun 27 2015
%C Definition of function F[a(n); b(n)]: Let a(n) and b(n) is pair of complements of natural numbers (A000027) with a(1) < a(2) < a(3) < ... and b(1) < b(2) < b(3) < ..., then F[a(n); b(n)] = F[a(n)] + F[b(n)]; where F[a(n)] = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... and F[b(n)] = 1/b(1) + 1/b(1)b(2) + 1/b(1)b(2)b(3) + ...
%C Value of function F[a(n); b(n)] is real number c = a + b, where a = real number whose Engel expansion is sequence a(n) and b = real number whose Engel expansion is sequence b(n). See A006784 for definition of Engel expansion.
%C Example for a(n) = odd numbers (A005408) and b(n) = even numbers (A005843): c = 2.059407... = a + b, where a = 1.410686... (A060196) and b = 0.648721... (A019774 - 1).
%C Example for a(n) = nonprime numbers (A018252) and b(n) = primes (A000040): c = 2.002747... = a + b, where a = 1.297516... and b = 0.705230... (A064648).
%C Conjecture: there are no pairs of complements a(n) and b(n) such that F[a(n); b(n)] = 2.
%C e - 1 <= F[a(n); b(n)] <= sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) - 1.
%C 1.71828182... (A091131) <= F[a(n); b(n)] <= 2.05940740....
%H G. C. Greubel, <a href="/A227569/b227569.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H Googology Wiki, <a href="http://googology.wikia.com/wiki/Double_factorial">Double factorial</a>
%e 2.05940740534257614453947549923327861297725472633534020929971877980544281968...
%t RealDigits[Sqrt[E] -1 + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
%t RealDigits[Sum[1/n!!, {n, 125}], 10, 105][[1]] (* _Robert G. Wilson v_, Apr 09 2014 *)
%o (PARI) default(realprecision, 100); exp(1/2) - 1 + sqrt(exp(1)*Pi/2)*(1-erfc(1/sqrt(2))) \\ _G. C. Greubel_, Apr 01 2019
%o (Magma) SetDefaultRealField(RealField(112)); R:= RealField(); -1 + Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // _G. C. Greubel_, Apr 01 2019
%o (Sage) numerical_approx(-1 + exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # _G. C. Greubel_, Apr 01 2019
%Y Cf. A000027, A005408, A005843, A091131 (e-1), A006882 (n!!), A143280 (m(2)).
%K nonn,cons
%O 1,1
%A _Jaroslav Krizek_, Jul 16 2013
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