login
Decimal expansion of average deviation of the total number of prime factors.
11

%I #33 Mar 18 2024 03:31:01

%S 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 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 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 Decimal expansion of average deviation of the total number of prime factors.

%C Or, decimal expansion of constant B2 from the summatory function of the restricted divisor function.

%C 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 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.

%D József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, p. 155, Chapter V, 1) b).

%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MertensConstant.html">Mertens Constant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>.

%F Equals A077761 + A136141. - _Jean-François Alcover_, Sep 02 2015

%F 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 From _Amiram Eldar_, Mar 18 2024: (Start)

%F Equals gamma + Sum_{k>=2} phi(k) * log(zeta(k)) / k, where phi = A000010.

%F Equals gamma - Sum_{p prime} 1/(p-1)^2 + Sum_{k>=2} J_2(k) * log(zeta(k)) / k, where J_2 = A007434.

%F Both formulas are from Jakimczuk (2017). (End)

%e 1.03465388189743791161979429846463825467030798434438525450307...

%t 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 Cf. A000010, A001620, A007434, A077761, A086242, A136141.

%K nonn,cons

%O 1,3

%A _Eric W. Weisstein_, Sep 25 2003