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2 added 846 characters in body

My interpretation is that the ratio column is approaching 0, with unusual rapidity for this sort of problem, so in the long run no $\log \log$ terms. Note that, for any prime not printed, the final column must be even closer to 0 than nearby primes that are printed.

In short, if $R(p)$ is that I do not see how the number of primes up to complete Ramanujan's construction as in "superior highly composite numbers," as his recipe gives factorizations for world champion numbers, and her we $p-2$ that are stuck with primes. Maybe something can be done with factors of quadratic residues $p-1$ or \pmod p,$and if$p+1$but N(p)$ is the number of primes up to $p-2$ that are quadratic nonresidues $\pmod p,$ I suggest

$$\lim_{p \rightarrow \infty} \frac{R(p) \log p}{p} = \lim_{p \rightarrow \infty} \frac{N(p) \log p}{p} = \frac{1}{2}.$$

From David's comment, this is also the prediction of a wild guesscertain generalized Riemann Hypothesis.

p res non diff diff/(res + non) 56039 2744 2941 -197 -0.0346526 56333 2936 2775 161 0.0281912 59399 2902 3102 -200 -0.0333111 61333 2976 3193 -217 -0.0351759 65539 3354 3189 165 0.0252178 69833 3351 3571 -220 -0.0317827 71971 3652 3471 181 0.0254106 81197 4074 3872 202 0.0254216 85223 4038 4259 -221 -0.0266361 85669 4053 4285 -232 -0.0278244 88919 4188 4425 -237 -0.0275165 89591 4216 4458 -242 -0.0278995 89659 4454 4229 225 0.0259127 95989 4504 4747 -243 -0.0262674 p res non diff diff/(res + non)
1

This was fun. I did my usual experiment. For each new prime $p,$ I looked at the prime numbers from 2 to $p-2,$ counted these by Jacobi symbol as either res or non, then took the difference diff = res - non. Then I printed out a line if either diff took on a new world record negative value or a world record positive value. Finally I put a decimal value, diff/(res + non), where res + non is the total number of primes up to $p-2.$

My interpretation is that the ratio column is approaching 0, with unusual rapidity for this sort of problem, so in the long run no $\log \log$ terms. Note that, for any prime not printed, the final column must be even closer to 0 than nearby primes that are printed.

The bad news is that I do not see how to complete Ramanujan's construction as in "superior highly composite numbers," as his recipe gives factorizations for world champion numbers, and her we are stuck with primes. Maybe something can be done with factors of $p-1$ or $p+1$ but that is a wild guess.

phoebus:~/Cplusplus> ./prime_res
5     0     2    -2              -1
13     1     4    -3            -0.6
19     4     3     1        0.142857
37     3     8    -5       -0.454545
107    15    12     3        0.111111
113    11    18    -7       -0.241379
139    19    14     5        0.151515
163    11    26   -15       -0.405405
211    26    20     6        0.130435
317    37    28     9        0.138462
373    28    45   -17       -0.232877
571    59    45    14        0.134615
647    49    68   -19       -0.162393
911    66    89   -23       -0.148387
1013    92    77    15       0.0887574
1031    74    98   -24       -0.139535
1093    77   105   -28       -0.153846
1097   100    83    17       0.0928962
1487   102   133   -31       -0.131915
1553   131   113    18       0.0737705
1613   139   115    24       0.0944882
1741   119   151   -32       -0.118519
1871   126   159   -33       -0.115789
2029   135   172   -37       -0.120521
2179   177   149    28       0.0858896
2293   149   191   -42       -0.123529
2851   223   190    33       0.0799031
2971   235   193    42       0.0981308
3637   230   278   -48      -0.0944882
4957   303   359   -56      -0.0845921
5419   379   336    43       0.0601399
5879   358   415   -57      -0.0737387
5923   357   420   -63      -0.0810811
6211   427   380    47       0.0582404
7213   423   498   -75      -0.0814332
7219   491   431    60       0.0650759
8731   581   506    75       0.0689972
10357   596   674   -78      -0.0614173
10627   596   699  -103      -0.0795367
15451   945   859    86       0.0476718
17491  1054   958    96       0.0477137
18119   985  1089  -104      -0.0501446
18439  1002  1109  -107      -0.0506869
21739  1277  1161   116         0.04758
21839  1168  1280  -112      -0.0457516
22669  1204  1327  -123      -0.0485974
23251  1355  1237   118       0.0455247
24181  1281  1410  -129      -0.0479376
26701  1396  1532  -136      -0.0464481
28607  1487  1626  -139      -0.0446515
31253  1748  1620   128       0.0380048
34483  1765  1917  -152      -0.0412819
35491  1958  1819   139       0.0368017
35933  1980  1836   144       0.0377358
36373  1852  2006  -154      -0.0399171
39839  2013  2173  -160      -0.0382226
43117  2168  2336  -168      -0.0373002
52453  2581  2775  -194      -0.0362211