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Does OEIS sequence A059046 contain any odd squares $u^2$, with $\omega(u) \geq 2$ (where $\omega(x)$ is the number of distinct prime factors of $x$)?

Here are the first sixty-two terms:

A059046 - Numbers $n$ such that $\sigma(n)-n$ divides $n-1$.

$2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211$

So, for example: $$\sigma({3^2}{5^2}) - 225 = {13}\cdot{31} - 225 = 403 - 225 = 178 \nmid 224,$$ so that $225$ is not in the sequence.

Without the constraint on the number of distinct prime factors $\omega(u')$, $u' = 9, 25, 49, 81, 121, 169, \ldots$.

OEIS sequence A059046

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    $\begingroup$ What is the motivation for this question ? $\endgroup$
    – Shailesh
    Oct 6, 2015 at 0:14
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    $\begingroup$ It's worth including the definition of the sequence - the image is hard for my poor eyes to read. $\endgroup$ Oct 6, 2015 at 0:15
  • $\begingroup$ @ThomasAndrews: "Numbers n such that sigma(n)-n divides n-1" for example "For x=77, sigma(77)=96, 96-77=19, which divides 77-1." $\endgroup$
    – Henry
    Oct 6, 2015 at 0:24
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    $\begingroup$ Put it in the question, @Henry. People should not have to read comments to know the question. And, except in rare occasions, people should not have to follow links. $\endgroup$ Oct 6, 2015 at 0:26
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    $\begingroup$ The squares upto $33,000,001^2$ with the desired property are not in the sequence. $\endgroup$
    – Peter
    Oct 6, 2015 at 22:42

1 Answer 1

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I checked that there are no such squares with $u^2<10^{12}$ using this simple GP script:

is(n)=(n-1)%(sigma(n)-n)==0
forcomposite(n=6,1e6,if(is(n^2)&&!isprimepower(n), print(n)))

I then used A059047 to check that there are no such squares with $u^2<10^{22}.$ Total computing time was about 12 seconds.

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  • $\begingroup$ Thank you for checking and for your answer, @Charles. I will wait for a couple more days and will accept your answer if there are no others. =) $\endgroup$ Oct 22, 2015 at 10:37
  • $\begingroup$ Such kind of casual proofs are disregarded FYI $\endgroup$
    – Abr001am
    Nov 25, 2017 at 18:14
  • $\begingroup$ @Idle I’m not sure what you mean, would you clarify? $\endgroup$
    – Charles
    Nov 26, 2017 at 2:59

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