Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

90 - The Only Deficiently Perfect Imperfect Number ?

  1. Jul 15, 2012 #1
    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. jcsd
  3. Jul 16, 2012 #2
    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: 90 - The Only Deficiently Perfect Imperfect Number ?
  1. Perfect Numbers (Replies: 18)

  2. Perfect Numbers (Replies: 5)

  3. Perfect number (Replies: 6)

Loading...