cp's OEIS Frontend

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

%I A138312 #33 Feb 16 2025 08:33:07
%S A138312 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,
%T A138312 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,
%U A138312 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
%N A138312 Decimal expansion of Mertens's constant B_3 minus Euler's constant.
%C A138312 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.
%C A138312 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.
%C A138312 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
%D A138312 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.
%H A138312 Robert G. Wilson v, <a href="/A138312/b138312.txt">Table of n, a(n) for n = 0..9999</a>
%H A138312 Dick Boland, <a href="http://www.imathination.org/kappa/kappa.pdf">An Analog of the Harmonic Numbers Over The Squarefree Integers</a>
%H A138312 David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/cohenb3.txt">The Mertens Constant</a>
%H A138312 Jean-Marie De Koninck and Nicolas Doyon, <a href="https://doi.org/10.1007/s00605-002-0493-0">À propos de l’indice de composition des nombres</a>, Monatshefte für Mathematik, Vol. 139, No. 2 (2003), pp. 151-167, <a href="http://www.jeanmariedekoninck.mat.ulaval.ca/fileadmin/jmdk/Documents/Publications/2003_a_propos_de_l_indice_de_composition_des_nombres.pdf">alternative link</a>.
%H A138312 Anne-Maria Ernvall-Hytönen, Tapani Matala-aho, and Louna Seppälä, <a href="https://arxiv.org/abs/1704.01374">On Mahler's transcendence measure for e</a>, arXiv:1704.01374 [math.NT], 2017. See Theorem 6.4.
%H A138312 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MertensConstant.html">Mertens Constant</a>
%H A138312 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%F A138312 Sum_{i>=1} log p_i/(p_i(p_i-1)), where p_i is the i-th prime.
%F A138312 Sum_{j>=2} mu(j)zeta'(j)/zeta(j), mu(j) is the Mobius function, zeta'(j) is the derivative of zeta(j).
%e A138312 0.755366610831688021159316685988625317796300153102499062981363664872472...
%t A138312 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 *)
%Y A138312 Cf. A083343 (Mertens' B_3), A001620 (Euler's Constant), A138313 (The constant 'Kappa' conjectured to be equivalent to this sequence), A138316, A138317, A007947.
%K A138312 cons,nonn
%O A138312 0,1
%A A138312 Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 14 2008, Mar 27 2008
%E A138312 More terms from _Jean-François Alcover_, Feb 14 2013