

A138312


Decimal expansion of Mertens's constant B_3 minus Euler's constant.


12



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

0,1


COMMENTS

Arises in the coefficients of the formula for the variance of the average order of omega(n), where omega(n) is the number of distinct prime factors of n  see MathWorld "Distinct Prime Factors" link and Hardy and Wright reference.
Conjectured to be equivalent to 'kappa' = limit_{n > infty)((sum_(k = 1..n) mu^2(k)/phi(k))  H_n), where mu(k) is the Mobius function, phi(k) is Euler's Totient and H_n is the nth harmonic number.
De Koninck and Doyon proved that the asymptotic sum of the index of composition Sum_{k<=x} log(k)/log(rad(k)) = x + c*x/log(x) + O(x/(log(x))^2), where c is this constant and rad(n) in the squarefree kernel of n (A007947).  Amiram Eldar, May 02 2019


REFERENCES

Hardy, G. H. and Wright, E. M., "The Number of Prime Factors of n" and "The Normal Order of omega(n) and Omega(n)." Sections 22.10 and 22.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 354358, 1979.


LINKS



FORMULA

Sum_{i>=1} log p_i/(p_i(p_i1)), where p_i is the ith prime.
Sum_{j>=2} mu(j)zeta'(j)/zeta(j), mu(j) is the Mobius function, zeta'(j) is the derivative of zeta(j).


EXAMPLE

0.755366610831688021159316685988625317796300153102499062981363664872472...


MATHEMATICA

f[n_] := f[n] = Sum[MoebiusMu[j]* Zeta'[j]/Zeta[j], {j, 2, n}] // RealDigits[#, 10, 105]& // First; f[100]; f[n = 200]; While[f[n] != f[n  100], n = n + 100]; f[n] (* JeanFrançois Alcover, Feb 14 2013, from 2nd formula *)


CROSSREFS



KEYWORD



AUTHOR

Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 14 2008, Mar 27 2008


EXTENSIONS



STATUS

approved



