1842675848 has four representations as the sum of two squares: 1842675848 = 3518^2 + 42782^2 = 5458^2 + 42578^2 = 11338^2 + 41402^2 = 19702^2 + 38138^2.
It got me to wonder which numbers in this range (say, less than 5 billion) have four or more representations as the sum of two squares? And which of these has the most?
Is there any systematic way to answer this question?
On
There is a procedure initiated by Ramanujan that can be applied to this problem. With the same definitions as those for Superior Highly Composite Numbers, here are the first few numbers that are "superior" in the count as the sum of two squares, where the numbers being squared are positive and in order.
The nature of the construction is that these numbers are quick to create, while each is guaranteed to have more representations (as the sum of two squares) than any smaller number.
Well, why not. We fix a real number $\delta > 0.$ For each prime $p \equiv 1 \pmod 4,$ we take the exponent $k$ of $p$ as $$ k = \left\lfloor \frac{1}{p^\delta - 1} \right\rfloor $$ If $p > 2^{1/\delta},$ then the denominator above exceeds $1$ and the relevant exponent is actually zero. That is, the product is actually a finite product.
Meanwhile, we are able to invert this in closed form. The first (largest) $\delta$ for which prime $p$ is given exponent $k$ is $$ \delta = \frac{\log \left( 1 + \frac{1}{k} \right)}{ \log p} $$
As in the srvey article by Nicolas, in a log-log graph of $\log n$ and $\log r_2(n),$ these "superior" values, in output below, show up as the vertices of the convex hull of the (infinite) graph. Note that I told it to multiply the vertical component by 4, to create a little vertical separation between nearby points. I will need to experiment to draw in the line segments, I may need to do that by hand. You can see, though, how some points in the plot below are higher compared with those nearby... Alright, added line segments and labels by hand
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Alright, I made it much faster, here are the first 54 such numbers. I wanted the list to reach both $5^5$ and $13^3,$ both of which have happened by the time we have reached $\delta = \frac{1}{9}.$
jagy@phobeusjunior:~$ ./Superior_Highly_Composite_read
1 5 = 5 log base ten 0.69897
2 65 = 5 13 log base ten 1.81291
3 325 = 5^2 13 log base ten 2.51188
6 5525 = 5^2 13 17 log base ten 3.74233
12 160225 = 5^2 13 17 29 log base ten 5.20473
24 5928325 = 5^2 13 17 29 37 log base ten 6.77293
48 243061325 = 5^2 13 17 29 37 41 log base ten 8.38572
64 1215306625 = 5^3 13 17 29 37 41 log base ten 9.08469
128 64411251125 = 5^3 13 17 29 37 41 53 log base ten 10.809
256 3929086318625 = 5^3 13 17 29 37 41 53 61 log base ten 12.5943
512 286823301259625 = 5^3 13 17 29 37 41 53 61 73 log base ten 14.4576
768 3728702916375125 = 5^3 13^2 17 29 37 41 53 61 73 log base ten 15.5716
1536 331854559557386125 = 5^3 13^2 17 29 37 41 53 61 73 89 log base ten 17.5209
3072 32189892277066454125 = 5^3 13^2 17 29 37 41 53 61 73 89 97 log base ten 19.5077
6144 3251179119983711866625 = 5^3 13^2 17 29 37 41 53 61 73 89 97 101 log base ten 21.512
12288 354378524078224593462125 = 5^3 13^2 17 29 37 41 53 61 73 89 97 101 109 log base ten 23.5495
24576 40044773220839379061220125 = 5^3 13^2 17 29 37 41 53 61 73 89 97 101 109 113 log base ten 25.6025
36864 680761144754269444040742125 = 5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 log base ten 26.833
73728 93264276831334913833581671125 = 5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 log base ten 28.9697
92160 466321384156674569167908355625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 log base ten 29.6687
184320 69481886239344510806018344988125 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 log base ten 31.8419
368640 10908656139577088196544880163135625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 log base ten 34.0378
737280 1887197512146836258002264268222463125 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 log base ten 36.2758
1474560 341582749698577362698409832548265825625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 log base ten 38.5335
2949120 65925470691825431000793097681815304345625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 log base ten 40.8191
5898240 12987317726289609907156240243317614956088125 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 log base ten 43.1135
11796480 2974095759320320668738779015719733824944180625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 log base ten 45.4734
23592960 692964311921634715816135510662697981211994085625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 log base ten 47.8407
47185920 167004399173113966511688658069710213472090574635625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 log base ten 50.2227
94371840 42920130587490289393503985123915524862327277681355625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 log base ten 52.6327
188743680 11545515128034887846852571998333276187966037696284663125 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 log base ten 55.0624
377487360 3198107690465663933578162443538317504066592441870851685625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 log base ten 57.5049
754974720 898668261020851565335463646634267218642712476165709323660625 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 log base ten 59.9536
1509949440 263309800479109508643290848463840295062314755516552831832563125 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 log base ten 62.4205
3019898880 82415967549961276205350035569182012354504518476681036363592258125 = 5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 log base ten 64.916
4529848320 2390063058948877009955151031506278358280631035823750054544175485625 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 log base ten 66.3784
9059696640 757649989686794012155782876987490239574960038356128767290503628943125 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 log base ten 68.8795
18119393280 255328046524449582096498829544784210736761532926015394576899722953833125 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 log base ten 71.4071
36238786560 89109488237032904151678091511129689547129774991179372707338003310887760625 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 log base ten 73.9499
72477573120 31455649347672615165542366303428780410136810571886318565690315168743379500625 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 log base ten 76.4977
144955146240 11732957206681885456747302631178935092981030343313596825002487557941280553733125 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 log base ten 79.0694
289910292480 4564120353399253442674700723528605751169620803548989164925967660039158135402185625 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 log base ten 81.6594
579820584960 1811955780299503616741856187240856483214339459008948698475609161035545779754667693125 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 log base ten 84.2581
1159641169920 726594267900100950313484331083583449768950123062588428088719273575253857681621744943125 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 log base ten 86.8613
2319282339840 297177055571141288678215091413185630955500600332598667088286182892278827791783293681738125 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 log base ten 89.473
4638564679680 125111540395450482533528553484951150632265752740024038844168482997649386500340766640011750625 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 log base ten 92.0973
9277129359360 54173296991230058937017863658983848223771070936430408819524953137982184354647551955125088020625 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 log base ten 94.7338
18554258718720 24323810349062296462721020782883747852473210850457253559966703958954000775236750827851164521260625 = 5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 log base ten 97.386
22265110462464 121619051745311482313605103914418739262366054252286267799833519794770003876183754139255822606303125 = 5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 log base ten 98.085
44530220924928 55579906647607347417317532488889363842901286793294824384523918546209891771415975641639910931080528125 = 5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 log base ten 100.745
89060441849856 25622336964546987159383382477377996731577493211708914041265526449802760106622764770795998939228123465625 = 5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 log base ten 103.409
133590662774784 948026467688238524897185151662985879068367248833229819526824478642702123945042296519451960751440568228125 = 5^5 13^2 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 log base ten 104.977
178120883699712 12324344079947100823663406971618816427888774234831987653848718222355127611285549854752875489768727386965625 = 5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 log base ten 106.091
356241767399424 6273091136693074319244674148553977561795386085529481715808997575178759954144344876069213624292282239965503125 = 5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 log base ten 108.797
jagy@phobeusjunior:~$
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Friday: I extended the table Gerry pointed out, up to $10^{15}$; athough I dropped the initial $2, 50$ as multiplying by $2$ does not change the number of representations ( I use $5$ and $65$). I give the number of representations (as sum of two squares, numbers that are to be squared taken as positive and then ordered), the factorization of the number, and its log base ten.
http://oeis.org/A007511
as text file: http://oeis.org/A007511/b007511.txt
The best you can do is $$ N = 5^{a_5} 13^{a_{13}} 17^{a_{17}}... p_r^{a_{p_r}} $$ with $$ a_5 \geq a_{13} \geq \cdots \geq a_{p_r} \geq 0 $$ and each prime $$ p_j \equiv 1 \pmod 4. $$
From the example below, it is clear we can take $p_r \leq 53.$
So, for example, you can do fairly well with all exponents $1$ and the product as large as possible below your bound, but you can do a little better.
To start, try $$ 5 \cdot 13 \cdot 17 \cdot 29 \cdot 37 \cdot 41 \cdot 53 = 2576450045 $$ Better, $$ 5^2 \cdot 13^2 \cdot 17 \cdot 29 \cdot 37 \cdot 41 = 3159797225 $$