# Does OEIS sequence A059046 contain any odd squares $u^2$, with $\omega(u) \geq 2$?

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 $$\omega(u')$$, $$u' = 9, 25, 49, 81, 121, 169, \ldots$$.

• What is the motivation for this question ? Oct 6, 2015 at 0:14
• It's worth including the definition of the sequence - the image is hard for my poor eyes to read. Oct 6, 2015 at 0:15
• @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." Oct 6, 2015 at 0:24
• 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. Oct 6, 2015 at 0:26
• The squares upto $33,000,001^2$ with the desired property are not in the sequence. Oct 6, 2015 at 22:42

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

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