A327824 Decimal expansion of the constant factor in the asymptotic for practical numbers (A005153).

1, 3, 3, 6, 0, 7

Offset

1,2

Comments

3 <= a(7) <= 7.

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))).

Margenstern evaluated it as 1.341.

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

Links

Table of n, a(n) for n=1..6.

Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, Journal of Number Theory 37 (1): 1-36, 1991.

Andreas Weingartner, Practical numbers and the distribution of divisors, Q. J. Math. 66 (2015), 743 - 758.

Andreas Weingartner, On the constant factor in several related asymptotic estimates, Mathematics of Computation, Vol. 88, No. 318 (2019), pp. 1883-1902. arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.

Andreas Weingartner, The constant factor in the asymptotic for practical numbers, arXiv:1906.07819 [math.NT], 2019.

Formula

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).

Example

1.33607...

Crossrefs

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

Sequence in context: A279062 A367763 A005882 * A189915 A085572 A342512

Adjacent sequences: A327821 A327822 A327823 * A327825 A327826 A327827

Keyword

nonn,cons,more

Author

Amiram Eldar, Sep 26 2019

Status

approved