A138312 Decimal expansion of Mertens's constant B_3 minus Euler's constant.

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

Offset: 0

Author

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

Keywords

• cons
• nonn

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' = lim_{n->oo} ((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 n-th 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

Examples

			0.755366610831688021159316685988625317796300153102499062981363664872472...
		

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. 354-358, 1979.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..9999
• Dick Boland, An Analog of the Harmonic Numbers Over The Squarefree Integers
• David Broadhurst, The Mertens Constant
• Jean-Marie De Koninck and Nicolas Doyon, À propos de l’indice de composition des nombres, Monatshefte für Mathematik, Vol. 139, No. 2 (2003), pp. 151-167, alternative link.
• Anne-Maria Ernvall-Hytönen, Tapani Matala-aho, and Louna Seppälä, On Mahler's transcendence measure for e, arXiv:1704.01374 [math.NT], 2017. See Theorem 6.4.
• Eric Weisstein's World of Mathematics, Mertens Constant
• Eric Weisstein's World of Mathematics, Distinct Prime Factors

Crossrefs

Cf. A083343 (Mertens' B_3), A001620 (Euler's Constant), A138313 (The constant 'Kappa' conjectured to be equivalent to this sequence), A138316, A138317, A007947.

Programs

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] (* Jean-François Alcover, Feb 14 2013, from 2nd formula *)

Formula

Sum_{i>=1} log p_i/(p_i(p_i-1)), where p_i is the i-th prime.

Sum_{j>=2} mu(j)zeta'(j)/zeta(j), mu(j) is the Mobius function, zeta'(j) is the derivative of zeta(j).

Extensions

More terms from Jean-François Alcover, Feb 14 2013