A047728 Intersection of A046985 and A033950: multiply perfect, refactorable numbers with integer average divisor dividing the number.

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 13661860101120, 154345556085770649600, 143573364313605309726720, 352338107624535891640320, 680489641226538823680000, 34384125938411324962897920, 156036748944739017459105792, 3638193973609385308194865152

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Colton proves that perfect numbers (A000396) cannot be refactorable.

Examples

			k = 45532800 is a term since s0 = d(k) = 384, s1 = sigma(k) = 571963392, and the four quotients s1/s0 = 474300, (k * s0)/s1 = 96, s1/k = 4 and k/s0 = 118580 are all integers.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..305
• Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

Crossrefs

Cf. A000005, A000203, A000396, A046985, A001599, A007691.

Programs

Mathematica

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n * d, s] && Divisible[s, d] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

PARI

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !((k * d) % s) && !(s % d) && !(k % d);} \\ Amiram Eldar, May 09 2024

Formula

Let s1 = sigma(k) = A000203(k) be the sum of divisors of k and s0 = d(k) = A000005(k) be the number of divisors of k. Then, k is a term if s1/s0, (k * s0)/s1, s1/k, and k/s0 are all integers.

Extensions

a(7)-a(13) from Donovan Johnson, Apr 09 2010

Edited and a(14)-a(16) added by Amiram Eldar, May 09 2024