A083342 Decimal expansion of average deviation of the total number of prime factors.

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, 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, 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

Offset: 1

Author

Eric W. Weisstein, Sep 25 2003

Keywords

• nonn
• cons

Comments

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

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

Examples

			1.03465388189743791161979429846463825467030798434438525450307...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.
• József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, p. 155, Chapter V, 1) b).

Links

• Rafael Jakimczuk, On Sums of Powers of the p-adic Valuation of n!, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.6.
• Dimbinaina Ralaivaosaona and Faratiana Brice Razakarinoro, An explicit upper bound for Siegel zeros of imaginary quadratic fields, arXiv:2001.05782 [math.NT], 2020.
• Eric Weisstein's World of Mathematics, Mertens Constant.
• Eric Weisstein's World of Mathematics, Prime Factor.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

Equals A077761 + A136141. - Jean-François Alcover, Sep 02 2015

Equals gamma + Sum_{p prime} (log(1-1/p) + 1/(p-1)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021

From Amiram Eldar, Mar 18 2024: (Start)

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

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

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