A047728 - OEIS

%I #18 May 09 2024 02:56:09

%S 1,672,30240,23569920,45532800,14182439040,153003540480,403031236608,

%T 13661860101120,154345556085770649600,143573364313605309726720,

%U 352338107624535891640320,680489641226538823680000,34384125938411324962897920,156036748944739017459105792,3638193973609385308194865152

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

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

%H Amiram Eldar, <a href="/A047728/b047728.txt">Table of n, a(n) for n = 1..305</a>

%H Simon Colton, <a href="http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

%F 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.

%e 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.

%t 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 *)

%o (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

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

%K nonn

%O 1,2

%A _Labos Elemer_

%E a(7)-a(13) from _Donovan Johnson_, Apr 09 2010

%E Edited and a(14)-a(16) added by _Amiram Eldar_, May 09 2024