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

1,2

Comments

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

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.

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.

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A340864 A331666 A245782 * A297123 A335254 A160209

Adjacent sequences: A047725 A047726 A047727 * A047729 A047730 A047731

Keyword

nonn,changed

Author

Labos Elemer

Extensions

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

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

Status

approved