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A234614 Decimal expansion of constant related to the growth of the number of totients. 4

%I

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

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

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

%N Decimal expansion of constant related to the growth of the number of totients.

%C Let f_k(x) = x * exp(k (log log log x)^2)/log x. Maier & Pomerance show that, for any e > 0, f_{c-e}(x) << g(x) << f_{c+e}(x) where g(x) gives the number of totients less than x and c is this constant. Loosely, this means f_c(A007617(n)) is about n.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 16.

%H Kevin Ford, <a href="http://www.math.uiuc.edu/~ford/wwwpapers/totients.pdf">The distribution of Totients</a>

%H Helmut Maier and Carl Pomerance, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4934.pdf">On the number of distinct values of Euler's phi-function</a>, Acta Arithmetica 49 (1988), pp. 263-275.

%F See Maier & Pomerance p. 264.

%F Equals -1/(2*log(c0)), where c0 is a constant whose decimal expansion is A246746. - _Amiram Eldar_, Jun 19 2018

%e 0.81781464008363223152559680090296560386485298237899...

%t digits = 101; F[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*x^k, {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; rho = x /. FindRoot[F[x] == 1, {x, 5/10, 6/10}, WorkingPrecision -> digits + 10]; RealDigits[rho, 10, digits] // First ;RealDigits[-1/2/Log[rho],10,90][[1]] (* after _Jean-Fran├žois Alcover_ at A246746 *)

%Y Cf. A007617, A246746.

%K nonn,cons

%O 0,1

%A _Charles R Greathouse IV_, Dec 28 2013

%E a(8) corrected and more terms added by _Amiram Eldar_, Jun 19 2018

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Last modified August 7 17:38 EDT 2020. Contains 336278 sequences. (Running on oeis4.)