# 90 - The Only Deficiently Perfect Imperfect Number ?

1. Jul 15, 2012

### Anti-Crackpot

90 - The Only "Deficiently Perfect Imperfect Number" ?

A125310 Numbers n such that n = sum of deficient proper divisors of n.
6, 28, 90, 496, 8128, 33550336
http://oeis.org/A125310

Joseph Pe offers the following comments:
COMMENTS 1. Since any proper divisor of an even perfect number is deficient, all even perfect numbers are (trivially) included in the sequence. 2. Hence the interesting terms of the sequence are its non-perfect terms, which I call "deficiently perfect". 90 is the only such term < 10^8.

And concludes with the following question:

"Are there any more?"

It's a question I share and I'm curious if anyone would care to extend the lower bound on this implied conjecture via brute force or offer a way to prove (or disprove) it outright and/or specify conditions such an integer would have to fulfill (such as, for instance, being abundant...)

TIA,
AC

Last edited: Jul 15, 2012
2. Jul 16, 2012

### Anti-Crackpot

Re: 90 - The Only "Deficiently Perfect Imperfect Number" ?

Here is one of a few very odd relationships from which this question stems...

Let...
(sigma_0(x) + sigma_1(x) + Phi(x)) = kx
(sigma_0(x') + sigma_1(x') + Phi(x')) = k'x

sigma_1|x - x'| = j|x - x'|
sigma_1|k - k'| = j'|k - k'|

x = 586, k = 2, j = 2
x' = 90, k' = 3, j' = 1
|x - x'| = 496 (a 2-Perfect Number)
|k - k'| = 1 (a 1-Perfect Number)

As 90 is the only "Deficiently Perfect Imperfect Number" < 10^8...

586 is the only n in N < 11*10^6 such that (sigma_0(x) + sigma_1(x) + Phi(x)) = 2x.
See: Numbers n such that n | Sigma(n) + d(n) + Phi(n)
1, 2, 4, 6, 90, 408, 586, 2200352, 11524640
http://oeis.org/A056012

What is the next one?

fwiw,
A) x + x', k+j and k'+j' are all Perfect Squares (26^2, 2^2, 2^2)
B) |x - x'|, |k - k'|, |j - j'| are all Triangular (T_31, T_1, T_1)

- AC

Last edited: Jul 16, 2012