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The title says it all.

What proportion of the positive integers satisfies $\gcd(n^2,\sigma(n^2))>\sigma(n)$, where $\sigma$ is the sum-of-divisors function?

I tried using Sage Cell Server to search for positive integers $n$ satisfying the said inequality, in the range $1 \leq n \leq {10}^6$, but it was unable to find any.

I dare to conjecture that the proportion in question is zero, but I have no proof.

Updated November 28 2018

I got the same output as Julian, after running the following (presumably correct) GP code in Sage Cell Server:

for (x=1, 1000000,if(gcd(x^2,sigma(x^2))>sigma(x),print(x)))

Updated November 29 2018

The number $n=1$ satisfies $$\gcd(n^2,\sigma(n^2))=\sigma(n)$$ while almost all integers $n>1$ (except for the $111$ numbers less than ${10}^6$ mentioned by Julian in his answer below) satisfy $$\gcd(n^2,\sigma(n^2))<\sigma(n).$$

Therefore, I think it might indeed be possible to prove that the set $$\bigg\{n \in \mathbb{N} \mid \gcd(n^2,\sigma(n^2)) > \sigma(n)\bigg\}$$ has asymptotic density $0$. But then again, as I have already mentioned, I have no proof.

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  • $\begingroup$ Note that $n$ must be composite. $\endgroup$ Nov 28, 2018 at 14:48
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    $\begingroup$ There are $111$ integers below $10^6$ satisfying the inequality. The smallest is $693$. $\endgroup$ Nov 28, 2018 at 15:01
  • $\begingroup$ Blimey! @Julian, please write out your comment as an actual answer so that I may be able to accept it. $\endgroup$ Nov 28, 2018 at 15:03
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    $\begingroup$ How often does it happen that $\gcd(n,\sigma(n^2))\gt1$? $\endgroup$ Nov 29, 2018 at 12:33
  • $\begingroup$ @BarryCipra, very often, I must say! Why do you ask? $\endgroup$ Nov 29, 2018 at 13:06

1 Answer 1

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Using Mathematica I found that there are $111$ integers below $10^6$ satisfying the inequality. They are

\begin{align} &693, 1386, 1463, 1881, 2379, 2926, 4389, 4758, 8778, 9516, 11895, \\ &13167, 16653, 18018, 19032, 23790, 24180, 25641, 26169, 26334, 33306, \\ &37271, 40443, 43890, 45201, 52668, 54717, 57057, 61380, 65835, 73150, \\ &78507, 105336, 109725, 111813, 114114, 131670, 157014, 166530, \\ &169959, 171171, 183183, 185801, 210672, 214830, 218085, 219450, \\ &223626, 223839, 230763, 233142, 238887, 250173, 263340, 291555, \\ &294996, 302841, 316407, 329175, 342342, 366366, 368745, 371602, \\ &380737, 381843, 392535, 405384, 408177, 421344, 429660, 447252, \\ &447678, 461526, 477774, 487179, 497211, 500346, 526680, 530439, \\ &539847, 549549, 557403, 559065, 566181, 570570, 605682, 611403, \\ &615660, 618849, 632814, 637032, 658350, 684684, 692289, 697851, \\ &737490, 763686, 766038, 776853, 782691, 803187, 809424, 816354, \\ &855855, 882189, 894504, 901641, 923052, 934857, 961191, 974358 \end{align} The sequence is now in OEIS.

Code:

Select[Range[10^6], GCD[#^2, DivisorSigma[1, #^2]] > DivisorSigma[1, #] &]
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