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

Author

Charles R Greathouse IV, Dec 28 2013

Keywords

• nonn
• cons

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.

Examples

			0.81781464008363223152559680090296560386485298237899...
		

Links

• 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.

Crossrefs

Cf. A007617, A246746.

Programs

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 *)

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

Extensions

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