A327832 - OEIS

%I #22 Dec 17 2024 06:30:23

%S 1,2,1,4,1,6,1,8,1,2,1,12,1,2,1,16,1,18,1,20,1,2,1,24,1,2,1,28,1,30,1,

%T 32,1,2,1,36,1,2,1,40,1,42,1,4,1,2,1,48,1,2,1,4,1,54,1,56,1,2,1,60,1,

%U 2,1,64,1,66,1,4,1,2,1,72,1,2,1,4,1,78,1,80,1

%N The practical component of n: the largest divisor of n which is a practical number (A005153).

%C From _Andreas Weingartner_, Jun 30 2021: (Start)

%C 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), ...

%C 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)

%H Amiram Eldar, <a href="/A327832/b327832.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Pollack and Lola Thompson, <a href="http://publi.math.unideb.hu/load_jpg.php?p=1792">Practical pretenders</a>, Publicationes Mathematicae Debrecen, Vol. 82, No. 3-4 (2013), pp. 651-717, <a href="http://arxiv.org/abs/1201.3168">arXiv preprint</a>, arXiv:1201.3168 [math.NT], 2012.

%H Andreas Weingartner, <a href="http://publi.math.unideb.hu/load_jpg.php?p=1996">Integers with large practical component</a>, Publicationes Mathematicae Debrecen, Vol. 87, No. 3-4 (2015), pp. 439-447, <a href="https://arxiv.org/abs/1411.6974">arXiv preprint</a>, arXiv:1411.6974v2 [math.NT], 2014-2015.

%F 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.

%F A number n is practical if and only if a(n) = n.

%F a(n) = 1 if and only if n is odd.

%F A000203(a(n)) = A225561(n).

%e a(22) = 2 since the divisors of 22 are {1, 2, 11, 22}, of them {1, 2} are practical, and 2 being the largest.

%t 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]

%o (PARI) \\ using is_A005153

%o a(n) = fordiv(n, d, if(is_A005153(n/d), return(n/d))); \\ _Michel Marcus_, Jul 03 2021

%Y Cf. A005153, A225561, A225574.

%K nonn

%O 1,2

%A _Amiram Eldar_, Sep 27 2019