

A327824


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


0




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: thÃ©orie, observations et conjectures, Journal of Number Theory 37 (1): 136, 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. 18831902. arXiv preprint, arXiv:1705.06349 [math.NT], 20172018.
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)/(p1)  log(k)) * Product_{p prime, p<=sigma(k)+1} (11/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: A265960 A279062 A005882 * A189915 A085572 A205548
Adjacent sequences: A327821 A327822 A327823 * A327825 A327826 A327827


KEYWORD

nonn,cons,more


AUTHOR

Amiram Eldar, Sep 26 2019


STATUS

approved



