On the equation involving the sum of divisors function $\sigma(105n+\sigma(n))=108\sigma(n)$

Question

I am curious about the solutions of the following equation involving the sum of divisors function $\sigma(m)=\sum_{d\mid m}d$

$$\sigma(105n+\sigma(n))=108\sigma(n).\tag{1}$$ It is obvious, since $107$ isn't a Mersenne prime, that every even perfect number is a solution of our equation $(1)$. And if we presume that there exist some odd perfect numbers coprime with the prime $107$, these should be also solutions of our equation.

Computational fact. Our sequence starts as $$6,28,402,496,1512,1710,1876,7980,8128,15012,29967,30267,\ldots$$ that you can see using Sage Cell Server (choose GP as language) with this code

for (i = 1, 1000000,if(sigma(105*i+sigma(i))==108*sigma(i),print(i)))

Question. Is it possible to prove that the equation $$\sigma(105n+\sigma(n))=108\sigma(n)$$ has infinitely many solutions? What work can be done*? Many thanks.

*Since I think that this is a difficult question (this kind of equations involving arithmetic functions are difficult, I should to accept an answer showing what work can be done, in the case that a full answer isn't feasible). Thus I am asking if you can to show an infinitude of solutions, or a compelling argument about why we can build such sequence.

Answer 1

Here is the list of all such $n<5000000$, which would be too long for a comment.

$n$:6, $sigma(n)+105n$: 642

$n$:28, $sigma(n)+105n$: 2996

$n$:402, $sigma(n)+105n$: 43026

$n$:496, $sigma(n)+105n$: 53072

$n$:1512, $sigma(n)+105n$: 163560

$n$:1710, $sigma(n)+105n$: 184230

$n$:1876, $sigma(n)+105n$: 200788

$n$:7980, $sigma(n)+105n$: 864780

$n$:8128, $sigma(n)+105n$: 869696

$n$:15012, $sigma(n)+105n$: 1615460

$n$:29967, $sigma(n)+105n$: 3192231

$n$:30267, $sigma(n)+105n$: 3226035

$n$:33232, $sigma(n)+105n$: 3556816

$n$:34344, $sigma(n)+105n$: 3704130

$n$:50844, $sigma(n)+105n$: 5464060

$n$:51912, $sigma(n)+105n$: 5613000

$n$:54405, $sigma(n)+105n$: 5820045

$n$:74028, $sigma(n)+105n$: 7952140

$n$:141360, $sigma(n)+105n$: 15318960

$n$:143295, $sigma(n)+105n$: 15281847

$n$:155610, $sigma(n)+105n$: 16863210

$n$:156450, $sigma(n)+105n$: 16873650

$n$:228510, $sigma(n)+105n$: 24587910

$n$:273084, $sigma(n)+105n$: 29402268

$n$:309582, $sigma(n)+105n$: 33377526

$n$:328524, $sigma(n)+105n$: 35371308

$n$:353400, $sigma(n)+105n$: 38297400

$n$:386508, $sigma(n)+105n$: 41515180

$n$:430350, $sigma(n)+105n$: 46317630

$n$:527436, $sigma(n)+105n$: 57123612

$n$:544576, $sigma(n)+105n$: 58285888

$n$:544908, $sigma(n)+105n$: 58783340

$n$:604359, $sigma(n)+105n$: 64479807

$n$:707310, $sigma(n)+105n$: 76176990

$n$:755524, $sigma(n)+105n$: 80998372

$n$:877770, $sigma(n)+105n$: 94507290

$n$:900135, $sigma(n)+105n$: 96099759

$n$:912285, $sigma(n)+105n$: 97624485

$n$:967176, $sigma(n)+105n$: 104735880

$n$:1055970, $sigma(n)+105n$: 113693490

$n$:1125252, $sigma(n)+105n$: 121241316

$n$:1272315, $sigma(n)+105n$: 135833859

$n$:1475010, $sigma(n)+105n$: 158859666

$n$:1505205, $sigma(n)+105n$: 160981821

$n$:1505286, $sigma(n)+105n$: 161339454

$n$:1524348, $sigma(n)+105n$: 164669148

$n$:1532640, $sigma(n)+105n$: 165959136

$n$:1654884, $sigma(n)+105n$: 179138820

$n$:1779996, $sigma(n)+105n$: 191410940

$n$:1886544, $sigma(n)+105n$: 204009360

$n$:2008300, $sigma(n)+105n$: 216148940

$n$:2175360, $sigma(n)+105n$: 236050560

$n$:2316480, $sigma(n)+105n$: 251033280

$n$:2329500, $sigma(n)+105n$: 251385372

$n$:2438004, $sigma(n)+105n$: 262523380

$n$:2678400, $sigma(n)+105n$: 291350400

$n$:3044734, $sigma(n)+105n$: 325731246

$n$:3259620, $sigma(n)+105n$: 354490500

$n$:3553950, $sigma(n)+105n$: 382985550

$n$:3822819, $sigma(n)+105n$: 408707355

$n$:3975450, $sigma(n)+105n$: 427868010

$n$:4418820, $sigma(n)+105n$: 480063780

$n$:4427280, $sigma(n)+105n$: 482738256

$n$:4452570, $sigma(n)+105n$: 480065130

$n$:4510302, $sigma(n)+105n$: 482602326

$n$:4567752, $sigma(n)+105n$: 495295560

$n$:4810428, $sigma(n)+105n$: 519909012

$n$:4837488, $sigma(n)+105n$: 520840176

$n$:4972968, $sigma(n)+105n$: 541515240

Answer 2

Let $$\mu(n) = \sigma(n)+105n,\tag1$$ $$\nu(m) = \dfrac1{105}\left(m-\dfrac{\sigma(m)}{108}\right),\tag2$$ then the issue equality can be presented in the form of $$\nu(\mu(n)) = n.\tag3$$ In particular, for perfect $n$ in the case $\gcd(n, 107) = 1$ $$\mu(n) = \sigma(n) + 105n = 107n,$$ $$\nu(\mu(n)) = \dfrac1{105}\left(107n - \dfrac{\sigma(107n)}{108}\right) = \dfrac1{105}\left(107n - \dfrac{(107+1)2n}{108}\right) = n.$$ Formula $(2)$ allows to look for possible values of $m=\mu(n)$ using the system $$108\ |\ \sigma(m),\quad 105\ |\ m - \dfrac {\sigma(m)}{108},\tag4 $$ and then verify them by the solving of the equation $$\sigma(n)+105n = m.$$

One can see that the possible solutions of the system $(4)$ can be presented in the forms of $$m = 108k-1 \in \mathbb P,\quad m = 107, 431, 641, 863, 971\dots\tag5$$ $$m = 85k,\quad \gcd(k, 85) = 1,\quad m = 85, 170, 255, 340, 510, 595\dots \tag 6$$ $$m = 102k,\quad \gcd(k, 102) = 1,\quad m=102, 510, 714, 1122\dots \tag 7$$ and some others ($m= 23k,47k, 71k$), according to the factorization of $108$ in sigma-functions.

This way can be helpful as some start ideas for analytic investigation of this hard topic.