Skip to main content
3 of 7
added 233 characters in body
Will Jagy
  • 25.7k
  • 2
  • 65
  • 121

I did want to see the behavior of specific forms of low discriminant from Nipp's tables, as Jeremy briefly indicated in an email. To get $r(n) \geq 15 \sigma(n)$ we seem to need discriminant $d \leq 21.$ To get $r(n) \geq 20 \sigma(n)$ we seem to need discriminant $d = 4,5.$

I should add that there are infinitely many forms that give $r(1) = 12,$ so that this ratio is at least $12.$ Given any positive integer $T \geq 2,$ $$ ( x^2 + y^2 + z^2 + yz + zx + xy) + T w^2 $$ represents $1$ twelve times.

Discriminant $4$ achieves the ratio $24.$ For $d=5,$ with prime $p \equiv \pm 2 \pmod 5,$ we get $$ r(p) = 30 (p-1) = 30 \; \sigma(p) \cdot \left( \frac{p-1}{p+1} \right) $$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

d = 5
                                          n      reps     sigma 
 ratio  20                                1       20        1  1 =  1 
 ratio  10                                2       30        3  2 = 2
 ratio  15                                3       60        4  3 = 3
 ratio  20                                5      120        6  5 = 5
 ratio  22.5                              7      180        8  7 = 7
 ratio  25.7142857142857                 13      360       14  13 = 13
 ratio  26.6666666666667                 17      480       18  17 = 17
 ratio  27.5                             23      660       24  23 = 23
 ratio  28.4210526315789                 37     1080       38  37 = 37
 ratio  28.6363636363636                 43     1260       44  43 = 43
 ratio  28.75                            47     1380       48  47 = 47
 ratio  28.8888888888889                 53     1560       54  53 = 53
 ratio  29.1176470588235                 67     1980       68  67 = 67
 ratio  29.1891891891892                 73     2160       74  73 = 73
 ratio  29.2857142857143                 83     2460       84  83 = 83
 ratio  29.3877551020408                 97     2880       98  97 = 97
 ratio  29.4230769230769                103     3060      104  103 = 103
 ratio  29.4444444444444                107     3180      108  107 = 107
 ratio  29.4736842105263                113     3360      114  113 = 113
 ratio  29.53125                        127     3780      128  127 = 127

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  d  4 record ratio  24 number  1       sigma  1   reps  24
  d  5 record ratio  29.5652173913043 number  137       sigma  138   reps  4080
  d  8 record ratio  17.76 number  149       sigma  150   reps  2664
  d  9 record ratio  12 number  1       sigma  1   reps  12
  d  12 record ratio  19.4117647058824 number  67       sigma  68   reps  1320
  d  12 record ratio  19.7333333333333 number  149       sigma  150   reps  2960
  d  13 record ratio  13.8133333333333 number  149       sigma  150   reps  2072
  d  16 record ratio  8 number  1       sigma  1   reps  8
  d  16 record ratio  12 number  1       sigma  1   reps  12
  d  17 record ratio  8.87142857142857 number  139       sigma  140   reps  1242
  d  20 record ratio  17.7391304347826 number  137       sigma  138   reps  2448
  d  20 record ratio  13.6216216216216 number  146       sigma  222   reps  3024
  d  20 record ratio  11.8260869565217 number  137       sigma  138   reps  1632
  d  21 record ratio  15.7866666666667 number  149       sigma  150   reps  2368
  d  21 record ratio  15.7714285714286 number  139       sigma  140   reps  2208
  d  24 record ratio  11.8125 number  127       sigma  128   reps  1512
  d  24 record ratio  12 number  1       sigma  1   reps  12
  d  24 record ratio  11.8260869565217 number  137       sigma  138   reps  1632
  d  25 record ratio  6 number  1       sigma  1   reps  6
  d  28 record ratio  9.80392156862745 number  101       sigma  102   reps  1000
  d  28 record ratio  12 number  1       sigma  1   reps  12
  d  28 record ratio  9.84375 number  127       sigma  128   reps  1260
  d  29 record ratio  10 number  5       sigma  6   reps  60
  d  29 record ratio  12 number  1       sigma  1   reps  12
  d  32 record ratio  9.86666666666667 number  149       sigma  150   reps  1480
  d  32 record ratio  11.84 number  149       sigma  150   reps  1776
  d  32 record ratio  12 number  1       sigma  1   reps  12
  d  32 record ratio  8.41176470588235 number  134       sigma  204   reps  1716
  d  32 record ratio  11.8285714285714 number  139       sigma  140   reps  1656
  d  33 record ratio  8 number  1       sigma  1   reps  8
  d  33 record ratio  7.88571428571429 number  139       sigma  140   reps  1104
  d  33 record ratio  7.88405797101449 number  137       sigma  138   reps  1088
  d  36 record ratio  12 number  1       sigma  1   reps  12
  d  36 record ratio  12 number  5       sigma  6   reps  72
  d  36 record ratio  8 number  1       sigma  1   reps  8
  d  36 record ratio  11.9504132231405 number  81       sigma  121   reps  1446
  d  36 record ratio  7.9843137254902 number  128       sigma  255   reps  2036
  d  37 record ratio  12 number  1       sigma  1   reps  12
  d  37 record ratio  7.5 number  3       sigma  4   reps  30
  d  40 record ratio  12 number  1       sigma  1   reps  12
  d  40 record ratio  7.71875 number  127       sigma  128   reps  988
  d  40 record ratio  7.92156862745098 number  101       sigma  102   reps  808
  d  40 record ratio  7.84 number  149       sigma  150   reps  1176
  d  41 record ratio  8 number  1       sigma  1   reps  8
  d  41 record ratio  5.33333333333333 number  2       sigma  3   reps  16
  d  44 record ratio  8.64 number  149       sigma  150   reps  1296
  d  44 record ratio  8.51612903225806 number  61       sigma  62   reps  528
  d  44 record ratio  12 number  1       sigma  1   reps  12
  d  44 record ratio  8.44444444444444 number  71       sigma  72   reps  608
  d  45 record ratio  11.8260869565217 number  137       sigma  138   reps  1632
  d  45 record ratio  12 number  1       sigma  1   reps  12
  d  45 record ratio  11.8125 number  127       sigma  128   reps  1512
  d  45 record ratio  11.5 number  141       sigma  192   reps  2208
  d  45 record ratio  11.8260869565217 number  137       sigma  138   reps  1632
  d  48 record ratio  11.84 number  149       sigma  150   reps  1776
  d  48 record ratio  9.86666666666667 number  149       sigma  150   reps  1480
  d  48 record ratio  9.86666666666667 number  149       sigma  150   reps  1480
  d  48 record ratio  12 number  1       sigma  1   reps  12
  d  48 record ratio  9.85714285714286 number  139       sigma  140   reps  1380
  d  48 record ratio  8.98765432098766 number  106       sigma  162   reps  1456
  d  48 record ratio  11.8285714285714 number  139       sigma  140   reps  1656
  d  48 record ratio  12 number  11       sigma  12   reps  144
  d  48 record ratio  9.05882352941176 number  134       sigma  204   reps  1848
  d  49 record ratio  4 number  1       sigma  1   reps  4
  d  52 record ratio  6.52941176470588 number  134       sigma  204   reps  1332
  d  52 record ratio  6.40740740740741 number  142       sigma  216   reps  1384
  d  52 record ratio  12 number  1       sigma  1   reps  12
  d  52 record ratio  8.3265306122449 number  97       sigma  98   reps  816
  d  52 record ratio  5.55102040816327 number  97       sigma  98   reps  544
  d  52 record ratio  6 number  1       sigma  1   reps  6
  d  53 record ratio  8.12903225806452 number  61       sigma  62   reps  504
  d  53 record ratio  12 number  1       sigma  1   reps  12
  d  53 record ratio  8 number  11       sigma  12   reps  96
  d  56 record ratio  8 number  1       sigma  1   reps  8
  d  56 record ratio  7.22222222222222 number  71       sigma  72   reps  520
  d  56 record ratio  7.46666666666667 number  89       sigma  90   reps  672
  d  56 record ratio  12 number  1       sigma  1   reps  12
  d  56 record ratio  7.15151515151515 number  131       sigma  132   reps  944
  d  57 record ratio  8 number  1       sigma  1   reps  8
  d  57 record ratio  5.66666666666667 number  149       sigma  150   reps  850
  d  57 record ratio  5.68421052631579 number  37       sigma  38   reps  216
  d  57 record ratio  6 number  1       sigma  1   reps  6
  d  60 record ratio  9.79591836734694 number  97       sigma  98   reps  960
  d  60 record ratio  12 number  1       sigma  1   reps  12
  d  60 record ratio  9.79591836734694 number  97       sigma  98   reps  960
  d  60 record ratio  9.86666666666667 number  149       sigma  150   reps  1480
  d  60 record ratio  9.86666666666667 number  149       sigma  150   reps  1480
  d  60 record ratio  9.85714285714286 number  139       sigma  140   reps  1380
Will Jagy
  • 25.7k
  • 2
  • 65
  • 121