A124175 Decimal expansion of Product_{primes p} ((p-1)/p)^(1/p).

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

Offset

0,1

Comments

This might be interpreted as the expected value of phi(n)/n for very large n.

Links

Table of n, a(n) for n=0..98.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 164.

Mathoverflow, Asymptotics of product of Euler's totient function, 2016.

Eric Weisstein's World of Mathematics, Prime Zeta Function

Formula

exp(-suminf(h=1, primezeta(h+1)/h)). - Robert Gerbicz

[Notation not clear. Is this perhaps exp(-Sum_{h=1..oo} primezeta(h+1)/h) ? - N. J. A. Sloane, Oct 08 2017]

Equals exp(1) * lim_{n->infinity} (A001088(n)/n!)^(1/n). - Vaclav Kotesovec, Feb 05 2016

Example

0.5598656169323734857237622442234167172576663702129060395542339339\
352031717975915936276540950630665470795373094197373037280781542375...

Mathematica

digits = 100; s = Exp[-NSum[PrimeZetaP[h+1]/h, {h, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 3 digits]]; RealDigits[s, 10, digits][[1]] (* Jean-François Alcover, Dec 07 2015, after Robert Gerbicz *)

Prog

(PARI) default(realprecision, 256); (f(k)=return(sum(n=1, 512, moebius(n)/n*log(zeta(k*n))))); exp(sum(h=1, 512, -1/h*f(h+1))) /* Robert Gerbicz */
(PARI) exp(-suminf(m=2, log(zeta(m))*sumdiv(m, k, if(k<m, moebius(k)/(m-k), 0)))) /* Martin Fuller */

Crossrefs

Cf. A126226, A085548, A085541, A085964 - A085969, A242624, A272028.

Sequence in context: A021951 A206772 A200679 * A168277 A163980 A333154

Adjacent sequences: A124172 A124173 A124174 * A124176 A124177 A124178

Keyword

nonn,cons

Author

David W. Wilson, Dec 05 2006

Extensions

Robert Gerbicz computed this to 130 decimal places.

Status

approved