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

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

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..86.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 16.

Kevin Ford, The distribution of Totients

Helmut Maier and Carl Pomerance, On the number of distinct values of Euler's phi-function, Acta Arithmetica 49 (1988), pp. 263-275.

Formula

See Maier & Pomerance p. 264.

Equals -1/(2*log(c0)), where c0 is a constant whose decimal expansion is A246746. - Amiram Eldar, Jun 19 2018

Example

0.81781464008363223152559680090296560386485298237899...

Mathematica

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-François Alcover at A246746 *)

Crossrefs

Cf. A007617, A246746.

Sequence in context: A200277 A242024 A159642 * A199872 A143548 A231929

Adjacent sequences: A234611 A234612 A234613 * A234615 A234616 A234617

Keyword

nonn,cons

Author

Charles R Greathouse IV, Dec 28 2013

Extensions

a(8) corrected and more terms added by Amiram Eldar, Jun 19 2018

Status

approved