A327832 The practical component of n: the largest divisor of n which is a practical number (A005153).

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

Offset: 1

Author

Amiram Eldar, Sep 27 2019

Keywords

• nonn

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)

Examples

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

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.

Crossrefs

Cf. A005153, A225561, A225574.

Programs

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]

PARI

\\ using is_A005153
a(n) = fordiv(n, d, if(is_A005153(n/d), return(n/d))); \\ Michel Marcus, Jul 03 2021

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.

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