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Decimal expansion of the constant factor in the asymptotic for practical numbers (A005153).
2

%I #9 Oct 21 2019 02:18:59

%S 1,3,3,6,0,7

%N Decimal expansion of the constant factor in the asymptotic for practical numbers (A005153).

%C 3 <= a(7) <= 7.

%C The constant c in the asymptotic function of the number of practical numbers up to x, P(x) = c*x/log(x) * (1 + O(log(log(x))/log(x))).

%C Margenstern evaluated it as 1.341.

%C Weingartner proved that 1.311 < c < 1.693 (2017), and 1.33607322 < c < 1.33607654 (2019).

%H Maurice Margenstern, <a href="http://dx.doi.org/10.1016/S0022-314X(05)80022-8">Les nombres pratiques: théorie, observations et conjectures</a>, Journal of Number Theory 37 (1): 1-36, 1991.

%H Andreas Weingartner, <a href="https://doi.org/10.1093/qmath/hav006">Practical numbers and the distribution of divisors</a>, Q. J. Math. 66 (2015), 743 - 758.

%H Andreas Weingartner, <a href="https://doi.org/10.1090/mcom/3402">On the constant factor in several related asymptotic estimates</a>, Mathematics of Computation, Vol. 88, No. 318 (2019), pp. 1883-1902. <a href="https://arxiv.org/abs/1705.06349">arXiv preprint</a>, arXiv:1705.06349 [math.NT], 2017-2018.

%H Andreas Weingartner, <a href="https://arxiv.org/abs/1906.07819">The constant factor in the asymptotic for practical numbers</a>, arXiv:1906.07819 [math.NT], 2019.

%F Equals 1/(1 - exp(-gamma)) * Sum_{k practical} (1/k) * (Sum_{p prime, p<=sigma(k)+1} log(p)/(p-1) - log(k)) * Product_{p prime, p<=sigma(k)+1} (1-1/p), where gamma is Euler's constant (A001620) and sigma is the divisors sum function (A000203).

%e 1.33607...

%Y Cf. A000203, A001620, A005153, A209237, A273773, A322257.

%K nonn,cons,more

%O 1,2

%A _Amiram Eldar_, Sep 26 2019