Define $\sigma(N)=\sum_{d|n} d$. A superabundant number is a positive integer $u$ for which $\frac{\sigma(u)}{u} > \frac{\sigma(v)}{v}$ for every positive integer $v<u$.
Similarly, do there exist infinitely many positive integers $n$ for which $\frac{\sigma(n^2)}{n^2}>\frac{\sigma(m^2)}{m^2}$ for every positive integer $m<n$ ?
The answer is yes because $\sigma(n^2)/n^2$ is unbounded. To see this, take the product of the first $k$ primes $n=p_1p_2\cdots p_k$. We have $$\frac{\sigma(n^2)}{n^2}=\prod_{i=1}^k \frac{p_i^{2}+p_i+1}{p_i^2}>\prod_{i=1}^k\left(1+\frac{1}{p_1}\right)>\sum_{i=1}^k\frac{1}{p_i}$$ and the final sum diverges as $k\to \infty$.
If not, some positive integer $n$ would maximize $\sigma(n^2)/n^2$. But if $m > 1$ and $n$ are coprime, $$\frac{\sigma((mn)^2)}{(mn)^2} = \frac{\sigma(m^2) \sigma(n^2)}{m^2 n^2} > \frac{\sigma(n^2)}{n^2}$$
Ramanujan's procedure, used in Alaoglu and Erdos, gives a best number for $\frac{\sigma(n^2)}{n^{2+\delta}}$ for any $0 < \delta < 1.$
For a fixed $\delta,$ the exponent of a prime $p$ is $$ \left\lfloor \frac{\log \left( p^{2 + \delta}-1 \right)-\log \left( p^{ \delta}-1 \right) - \log p }{2 \log p} \right\rfloor $$ For large $p$ the fraction is below $1$ and the floor is zero, so this is a finite factorization
For rational $\delta$ there is no ambiguity; $0.4$ gives $2;$ $0.2$ gives $6;$ $0.14$ gives $12;$ $0.1$ gives $60.$ $0.075$ gives $420.$Then $0.04$ gives $4620$ and $0.0344$ gives 9240.
Sometimes it is possible to invert the recipe above, for a given prime $p$ and desired exponent $k,$ find the first (largest) $\delta$ that assigns the proper exponent. If so, it becomes possible to write a simple program to put these in order. Not sure about this one....
Not that bad; I get $$ \delta = \frac{\log \left( p^{2 k+1}-1 \right)-\log \left( p^{ 2k+1}-p^2 \right) }{\log p} $$
Yes, that works. For a pair $(p,k)$ calculate $\delta$ and write out the line $$ \delta \hspace{10mm} p \hspace{10mm} k $$ as long as $\delta$ is bigger than some specified lower bound. Then sort the file by the $\delta$ lines. Finally, run the lines successively, keeping track of the product. At each new line, just multiply the cumulative product by that prime. I've done this for other multiplicative ratios.
jagy@phobeusjunior:~$ ./Superior_weird_ass_deltas 0.008
0.8073549220576041 2 1
0.1468413883292711 2 2
0.0344883763852905 2 3
0.0084947941654554 2 4
0.3347175194727928 3 1
0.0305991578154838 3 2
0.1336562149773228 5 1
0.0777173446560941 7 1
0.0394339918417963 11 1
0.0310288535275851 13 1
0.0213259829520153 17 1
0.0183129679967370 19 1
0.0141507419910474 23 1
0.0104090399683404 29 1
0.0095388401657691 31 1
===================================
jagy@phobeusjunior:~$ ./Superior_weird_ass_deltas 0.008 | sort -n -r
0.8073549220576041 2 1
0.3347175194727928 3 1
0.1468413883292711 2 2
0.1336562149773228 5 1
0.0777173446560941 7 1
0.0394339918417963 11 1
0.0344883763852905 2 3
0.0310288535275851 13 1
0.0305991578154838 3 2
0.0213259829520153 17 1
0.0183129679967370 19 1
0.0141507419910474 23 1
0.0104090399683404 29 1
0.0095388401657691 31 1
0.0084947941654554 2 4
This last, sorted, list tells us the primes to multiply the cumulative product.
Next morning: I ran things just to the point the ration $\frac{\sigma(n^2)}{n^2}$ is a little bit bigger than $10$
As either $n$ or $\sigma$ got too big I stopped printing them;
sigma(n^2) / n^2 sigma(n^2)... n = n factored
1.75 7 ... 2 = 2
2.527777777777777 91 ... 6 = 2 3
2.79861111111111 403 ... 12 = 2^2 3
3.470277777777778 12493 ... 60 = 2^2 3 5
4.036853741496595 712101 ... 420 = 2^2 3 5 7
4.437202872884701 94709433 ... 4620 = 2^2 3 5 7 11
4.544554555293197 388003161 ... 9240 = 2^3 3 5 7 11
4.921026530287906 71004578463 ... 120120 = 2^3 3 5 7 11 13
5.089266753545604 660888768771 ... 360360 = 2^3 3^2 5 7 11 13
5.406245305669577 202892852012697 ... 6126120 = 2^3 3^2 5 7 11 13 17
5.705760281052932 77302176616837557 ... 116396280 = 2^3 3^2 5 7 11 13 17 19
5.96462275127083 2677114440 = 2^3 3^2 5 7 11 13 17 19 23
6.177391696024804 77636318760 = 2^3 3^2 5 7 11 13 17 19 23 29
6.383090482989251 2406725881560 = 2^3 3^2 5 7 11 13 17 19 23 29 31
6.420785899227296 4813451763120 = 2^4 3^2 5 7 11 13 17 19 23 29 31
6.599010781747845 178097715235440 = 2^4 3^2 5 7 11 13 17 19 23 29 31 37
6.763887910143684 7302006324653040 = 2^4 3^2 5 7 11 13 17 19 23 29 31 37 41
6.924845761980521 313986271960080720 = 2^4 3^2 5 7 11 13 17 19 23 29 31 37 41 43
7.075317738700853 prime with bumped exponent 47
7.211333102848147 prime with bumped exponent 53
7.267162778483151 prime with bumped exponent 5
7.39242269422832 prime with bumped exponent 59
7.515596627859687 prime with bumped exponent 61
7.629443936991887 prime with bumped exponent 67
7.657467606181807 prime with bumped exponent 3
7.766838300021437 prime with bumped exponent 71
7.874690811600058 prime with bumped exponent 73
7.975632209601768 prime with bumped exponent 79
8.072881898324058 prime with bumped exponent 83
8.164607611093814 prime with bumped exponent 89
8.249646568037763 prime with bumped exponent 97
8.332134946622114 prime with bumped exponent 101
8.344364107502745 prime with bumped exponent 2
8.426163887611859 prime with bumped exponent 103
8.505649056610135 prime with bumped exponent 107
8.584398437657855 prime with bumped exponent 109
8.661038849741209 prime with bumped exponent 113
8.729772991521003 prime with bumped exponent 127
8.796921178391408 prime with bumped exponent 131
8.861600976069129 prime with bumped exponent 137
8.886983291790436 prime with bumped exponent 7
8.951378388361181 prime with bumped exponent 139
9.011857950464774 prime with bumped exponent 149
9.071934368536812 prime with bumped exponent 151
9.130085434715081 prime with bumped exponent 157
9.186441865566261 prime with bumped exponent 163
9.241779892509077 prime with bumped exponent 167
9.295509375662782 prime with bumped exponent 173
9.347729708598894 prime with bumped exponent 179
9.399659955141617 prime with bumped exponent 181
9.44913049354181 prime with bumped exponent 191
9.498343393639654 prime with bumped exponent 193
9.546803080617764 prime with bumped exponent 197
9.595018040243069 prime with bumped exponent 199
9.64070757606901 prime with bumped exponent 211
9.684133313506926 prime with bumped exponent 223
9.726982629338618 prime with bumped exponent 227
9.769644020324758 prime with bumped exponent 229
9.811753779221275 prime with bumped exponent 233
9.852978913708272 prime with bumped exponent 239
9.894032285673275 prime with bumped exponent 241
9.933607786633155 prime with bumped exponent 251
9.97241035455863 prime with bumped exponent 257
10.01047243921488 prime with bumped exponent 263
10.04782443212048 prime with bumped exponent 269
10.08503808996262 prime with bumped exponent 271
10.1215776068214 prime with bumped exponent 277
10.15772564047219 prime with bumped exponent 281
10.1937454944101 prime with bumped exponent 283
10.22865517507419 prime with bumped exponent 293
10.26208179810436 prime with bumped exponent 307
10.29518494551908 prime with bumped exponent 311
10.32818199635039 prime with bumped exponent 313
10.36086579134204 prime with bumped exponent 317
10.36466190728772 prime with bumped exponent 2
10.39606968700201 prime with bumped exponent 331
10.42701009815456 prime with bumped exponent 337
sigma(n^2) / n^2
