A046985 Multiply perfect numbers whose average divisor is an integer and divides the number itself.

1, 6, 672, 30240, 32760, 23569920, 45532800, 14182439040, 51001180160, 153003540480, 403031236608, 13661860101120, 154345556085770649600, 9186050031556349952000, 143573364313605309726720, 352338107624535891640320, 680489641226538823680000, 34384125938411324962897920

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..321

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

Example

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.

Mathematica

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 *)

Prog

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

Crossrefs

Intersection of A003601, A007691 and A001599.

Cf. A000005, A000203, A046986, A046987.

Sequence in context: A269842 A333639 A331724 * A159371 A159620 A125535

Adjacent sequences: A046982 A046983 A046984 * A046986 A046987 A046988

Keyword

nonn

Author

Labos Elemer

Extensions

a(10)-a(15) from Donovan Johnson, Nov 30 2008

Edited and a(16)-a(18) added by Amiram Eldar, May 09 2024

Status

approved