[seqfan] Re: A335071 question.
michel.marcus at free.fr
michel.marcus at free.fr
Sat May 30 23:14:02 CEST 2020
I had an idea like this:
Some relevant articles:
https://cs.uwaterloo.ca/journals/JIS/VOL10/Holdener/holdener7.pdf
https://www.researchgate.net/publication/265661827_Results_concerning_uniqueness_for_sxxsp_n_q_m_p_n_q_m_and_related_topics/link/5b258279458515270fd40f88/download
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-abundancy-ratio-a-measure-of-perfection
Using 1st article:
Property 2.2 If ab(n) = k/m with gcd(k, m) = 1, then m|n.
This follows directly from setting σ(n)/n = k/m.
Clearly, m|(nk) and since k and m are relatively prime, it must be that m|n.
Now if ab(n) = ab(n+1) = k/m with gcd(k, m) = 1, then m|n and also m|(n+1).
But n and n+1 are coprime. So it is not possible.
But there is a flaw there if they are multiperfect, that is with m=1, then m|n and m|(n+1) .
So ...
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