OFFSET
1,2
COMMENTS
From Andreas Weingartner, Jun 30 2021: (Start)
Let r_m be the natural density of the set of integers n with a(n) = m. Then r_m is positive if and only if m is practical. In that case, r_m = (1/m)*P_m, where P_m is the product of (1-1/p) over primes p <= sigma(m) + 1 (see Cor. 1 of Weingartner 2015). The first few values of (m, r_m) are (1, 1/2), (2, 1/6), (4, 2/35), (6, 32/1001), (8, 24/1001), (12, 36864/2800733), ...
As y grows, the natural density of integers n, which satisfy a(n) > y, is asymptotic to c*exp(-gamma)/log(y), where c = 1.33607... is the constant factor in the asymptotic for the count of practical numbers (A005153) and gamma = 0.577215... is Euler's constant (see Eq. (3) of Weingartner (2015)). For example, about 1% of integers n satisfy a(n) > exp(75), because c*exp(-gamma)/75 = 0.010... (End)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Lola Thompson, Practical pretenders, Publicationes Mathematicae Debrecen, Vol. 82, No. 3-4 (2013), pp. 651-717, arXiv preprint, arXiv:1201.3168 [math.NT], 2012.
Andreas Weingartner, Integers with large practical component, Publicationes Mathematicae Debrecen, Vol. 87, No. 3-4 (2015), pp. 439-447, arXiv preprint, arXiv:1411.6974v2 [math.NT], 2014-2015.
FORMULA
If n = Product_{i=1..r} p_i^e_i, then define n_0 = 1, n_j = Product_{i=1..j} p_i^e_i. a(n) = n_j where j is the first index for which p_{j+1} > sigma(n_j) + 1, or j = r if no such index exists.
A number n is practical if and only if a(n) = n.
a(n) = 1 if and only if n is odd.
EXAMPLE
a(22) = 2 since the divisors of 22 are {1, 2, 11, 22}, of them {1, 2} are practical, and 2 being the largest.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[n_] := If[(ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}, n, Times @@ (Power @@@ fct[[1 ;; ind[[1, 1]] - 1]])]; Array[a, 100]
PROG
(PARI) \\ using is_A005153
a(n) = fordiv(n, d, if (is_A005153(n/d), return(n/d)); \\ Michel Marcus, Jul 03 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 27 2019
STATUS
approved