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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 $m<n$ ?

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    $\begingroup$ The sequence $1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 2520, 4620, 9240, 13860, 27720, 55440, 60060, \ldots$ is not (yet) in the OEIS. $\endgroup$ – Robert Israel Jan 21 at 20:29
  • $\begingroup$ @RobertIsrael I worked out Ramanujan's sequence of "superior" numbers for this function; later called Colossal. Last time I tried to put something on the OEIS I got lost... I believe the previous one was the number of representations of an integer as the sum of two squares. $\endgroup$ – Will Jagy Jan 22 at 17:18
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    $\begingroup$ ... and now it is OEIS sequence A340816. $\endgroup$ – Robert Israel Jan 24 at 18:49
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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 $k\to \infty$.

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If not, some positive integer $n$ would maximize $\sigma(n^2)/n^2$. But if $m > 1$ and $n$ are coprime, $$\frac{\sigma((mn)^2)}{(mn)^2} = \frac{\sigma(m^2) \sigma(n^2)}{m^2 n^2} > \frac{\sigma(n^2)}{n^2}$$

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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    
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