This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A083342 #34 Feb 16 2025 08:32:49
%S A083342 1,0,3,4,6,5,3,8,8,1,8,9,7,4,3,7,9,1,1,6,1,9,7,9,4,2,9,8,4,6,4,6,3,8,
%T A083342 2,5,4,6,7,0,3,0,7,9,8,4,3,4,4,3,8,5,2,5,4,5,0,3,0,7,0,2,8,1,2,8,1,6,
%U A083342 3,3,5,3,9,3,8,6,6,0,1,6,0,7,5,4,7,9,4,1,3,9,0,2,5,7,5,6,7,4,6,9,3,8
%N A083342 Decimal expansion of average deviation of the total number of prime factors.
%C A083342 Or, decimal expansion of constant B2 from the summatory function of the restricted divisor function.
%C A083342 The constant A in the asymptotic formula Sum_{prime p <= n} 1/(p-1) = log(log(n)) + A + O(1/log(n)) (Jakimczuk, 2017). - _Amiram Eldar_, Mar 18 2024
%D A083342 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.
%D A083342 József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, p. 155, Chapter V, 1) b).
%H A083342 Rafael Jakimczuk, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Jakimczuk/jak37.html">On Sums of Powers of the p-adic Valuation of n!</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.6.
%H A083342 Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, <a href="https://arxiv.org/abs/2001.05782">An explicit upper bound for Siegel zeros of imaginary quadratic fields</a>, arXiv:2001.05782 [math.NT], 2020.
%H A083342 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MertensConstant.html">Mertens Constant</a>.
%H A083342 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>.
%F A083342 Equals A077761 + A136141. - _Jean-François Alcover_, Sep 02 2015
%F A083342 Equals gamma + Sum_{p prime} (log(1-1/p) + 1/(p-1)), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Dec 25 2021
%F A083342 From _Amiram Eldar_, Mar 18 2024: (Start)
%F A083342 Equals gamma + Sum_{k>=2} phi(k) * log(zeta(k)) / k, where phi = A000010.
%F A083342 Equals gamma - Sum_{p prime} 1/(p-1)^2 + Sum_{k>=2} J_2(k) * log(zeta(k)) / k, where J_2 = A007434.
%F A083342 Both formulas are from Jakimczuk (2017). (End)
%e A083342 1.03465388189743791161979429846463825467030798434438525450307...
%t A083342 digits = 102; Mp = EulerGamma - NSum[PrimeZetaP[n]/n - PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits]; RealDigits[Mp, 10, digits] // First (* _Jean-François Alcover_, Sep 02 2015 *)
%Y A083342 Cf. A000010, A001620, A007434, A077761, A086242, A136141.
%K A083342 nonn,cons
%O A083342 1,3
%A A083342 _Eric W. Weisstein_, Sep 25 2003