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 $d = 4,5.$ Then discriminant $5$ 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 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