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

1,2

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.

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

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

Cf. A005153, A225561, A225574.

Sequence in context: A328479 A340346 A193267 * A083258 A083259 A342242

Adjacent sequences: A327829 A327830 A327831 * A327833 A327834 A327835

Keyword

nonn

Author

Amiram Eldar, Sep 27 2019

Status

approved