A046985 - OEIS

%I #24 May 09 2024 02:48:10

%S 1,6,672,30240,32760,23569920,45532800,14182439040,51001180160,

%T 153003540480,403031236608,13661860101120,154345556085770649600,

%U 9186050031556349952000,143573364313605309726720,352338107624535891640320,680489641226538823680000,34384125938411324962897920

%N Multiply perfect numbers whose average divisor is an integer and divides the number itself.

%H Amiram Eldar, <a href="/A046985/b046985.txt">Table of n, a(n) for n = 1..321</a>

%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/k, (k * s0)/s1, and s1/s0 are all integers.

%e k = 45532800 is a term since, s0 = 384, s1 = 182131200, and the three quotients s1/k = 182131200/45532800 = 4, (k * s0)/s1 = (45532800 * 384)/182131200 = 96, and s1/s0 = 182131200/384 = 474300 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]]; Select[Range[33000], q] (* _Amiram Eldar_, May 09 2024 *)

%o (PARI) isok(n) = s1 = sigma(n); s0 = numdiv(n); !(s1 % n) && !(s1 % s0) && !((n*s0) % s1); \\ _Michel Marcus_, Dec 10 2013

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

%Y Intersection of A003601, A007691 and A001599.

%Y Cf. A000005, A000203, A046986, A046987.

%K nonn

%O 1,2

%A _Labos Elemer_

%E a(10)-a(15) from _Donovan Johnson_, Nov 30 2008

%E Edited and a(16)-a(18) added by _Amiram Eldar_, May 09 2024