

A234614


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


4



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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



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_{ce}(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 phifunction, Acta Arithmetica 49 (1988), pp. 263275.


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 JeanFrançois Alcover at A246746 *)


CROSSREFS

Cf. A007617, A246746.
Sequence in context: A200277 A242024 A159642 * A246750 A199872 A143548
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



