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List of superparticular intervals: Difference between revisions

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Consistency with other commas in the magnitude 10^-4.
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Completed the 29-limit section of the table.
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This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 101-limit and is complete up to the [[23-limit]].
This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 101-limit and is complete up to the [[29-limit]].


[[Wikipedia: Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than [[2/1]], [[3/2]], [[4/3]], and [[9/8]]. {{OEIS| A002071 }} gives the number of superparticular ratios in each prime limit, {{OEIS| A145604 }} shows the increment from limit to limit, and {{OEIS| A117581 }} gives the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).
[[Wikipedia: Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than [[2/1]], [[3/2]], [[4/3]], and [[9/8]]. {{OEIS| A002071 }} gives the number of superparticular ratios in each prime limit, {{OEIS| A145604 }} shows the increment from limit to limit, and {{OEIS| A117581 }} gives the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).
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<!--
1625/1624 -3 0 3 -1 0 1 0 0 0 -1
1683/1682 -1 2 0 0 1 0 1 0 0 -2
2001/2000 -4 1 -3 0 0 0 0 0 1 1
2002/2001 1 -1 0 1 1 1 0 0 -1 -1
2176/2175 7 -1 -2 0 0 0 1 0 0 -1
2205/2204 -2 2 1 2 0 0 0 -1 0 -1
2262/2261 1 1 0 -1 0 1 -1 -1 0 1
2465/2464 -5 0 1 -1 -1 0 1 0 0 1
2640/2639 4 1 1 -1 1 -1 0 0 0 -1
2755/2754 -1 -4 1 0 0 0 -1 1 0 1
2784/2783 5 1 0 0 -2 0 0 0 -1 1
3249/3248 -4 2 0 -1 0 0 0 2 0 -1
3451/3450 -1 -1 -2 1 0 0 1 0 -1 1
3510/3509 1 3 1 0 -2 1 0 0 0 -1
4641/4640 -5 1 -1 1 0 1 1 0 0 -1
4785/4784 -4 1 1 0 1 -1 0 0 -1 1
4901/4900 -2 0 -2 -2 0 2 0 0 0 1
5104/5103 4 -6 0 -1 1 0 0 0 0 1
5888/5887 8 0 0 -1 0 0 0 0 1 -2
5916/5915 2 1 -1 -1 0 -2 1 0 0 1
6670/6669 1 -3 1 0 0 -1 0 -1 1 1
7106/7105 1 0 -1 -2 1 0 1 1 0 -1
7425/7424 -8 3 2 0 1 0 0 0 0 -1
7889/7888 -4 0 0 3 0 0 -1 0 1 -1
8671/8670 -1 -1 -1 0 0 1 -2 0 1 1
9802/9801 1 -4 0 0 -2 2 0 0 0 1
10557/10556 -2 3 0 -1 0 -1 1 0 1 -1
11340/11339 2 4 1 1 0 0 -1 0 -1 -1
12006/12005 1 2 -1 -4 0 0 0 0 1 1
12673/12672 -7 -2 0 0 -1 0 0 1 1 1
13225/13224 -3 -1 2 0 0 0 0 -1 2 -1
13311/13310 -1 3 -1 0 -3 0 1 0 0 1
13312/13311 10 -3 0 0 0 1 -1 0 0 -1
-->
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|13456/13455
| 1625/1624
| 0.12866
| 1.0657
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| 2.3.5.13.23.29 {{monzo|4 -2 -1 -1 -1 2}}
| 2.5.7.13.29 {{monzo|-3 3 -1 1 -1}}
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| S116
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<!--
|-
19228/19227 2 -1 0 0 1 -1 -1 1 1 -1
| 1683/1682
20736/20735 8 4 -1 0 -1 -1 0 0 0 -1
| 1.0290
23751/23750 -1 2 -4 1 0 1 0 -1 0 1
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24795/24794 -1 2 1 -2 -1 0 0 1 -1 1
| 2.3.11.17.29 {{monzo|-1 2 1 1 -2}}
25840/25839 4 -4 1 0 -1 0 1 1 0 -1
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27000/26999 3 3 3 -2 0 0 0 -1 0 -1
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30625/30624 -5 -1 4 2 -1 0 0 0 0 -1
|-
30856/30855 3 -1 -1 1 -2 0 -1 1 0 1
| 2001/2000
35322/35321 1 1 0 1 -1 -2 0 -1 0 2
| 0.86540
47125/47124 -2 -2 3 -1 -1 1 -1 0 0 1
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53361/53360 -4 2 -1 2 2 0 0 0 -1 -1
| 2.3.5.23.29 {{monzo|-4 1 -3 1 1}}
72501/72500 -2 1 -4 0 1 3 0 0 0 -1
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83521/83520 -6 -2 -1 0 0 0 4 0 0 -1
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87465/87464 -3 1 1 3 0 -1 1 0 0 -2
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136851/136850 -1 1 -2 -1 2 1 -1 0 -1 1
| 2002/2001
158950/158949 1 -3 2 -1 1 0 2 0 0 -2
| 0.86497
166635/166634 -1 2 1 1 0 -2 -1 0 2 -1
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168751/168750 -1 -3 -5 0 1 0 0 0 2 1
| 2.3.7.11.13.23.29 {{monzo|1 -1 1 1 1 -1 -1}}
176001/176000 -7 1 -3 1 -1 0 2 0 0 1
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176176/176175 4 -5 -2 1 2 1 0 0 0 -1
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184093/184092 -2 -1 0 2 0 1 2 0 -2 -1
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240787/240786 -1 -3 0 -3 0 -1 0 2 1 1
| 2176/2175
244036/244035 2 -2 -1 0 -1 2 -1 2 0 -1
| 0.79579
-->
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| 2.3.5.17.29 {{monzo|7 -1 -2 1 -1}}
| 303601/303600
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| 0.0057023
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| 2.3.5.11.19.23.29 {{monzo|-4 -1 -2 -1 2 -1 2}}
| 2205/2204
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| 0.78532
| S551
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|-
| 2.3.5.7.19.29 {{monzo|-2 2 1 2 -1 -1}}
| 410670/410669
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| 0.0042156
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| 2.3.5.7.13.17.29 {{monzo|1 5 1 -2 2 -2 -1}}
| 2262/2261
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| 0.76552
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| 2.3.7.13.17.19.29 {{monzo|1 1 -1 1 -1 -1 1}}
| 418761/418760
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| 0.0041342
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| 2.3.5.7.17.19.23.29 {{monzo|-3 2 -1 1 2 -2 1 -1}}
| 2465/2464
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| 0.70247
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| 2.5.7.11.17.29 {{monzo|-5 1 -1 -1 1 1}}}
| 613089/613088
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| 0.0028238
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| 2640/2639
| 0.65589
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| 2.3.5.7.11.13.29 {{monzo|4 1 1 -1 1 -1 -1}}
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| 2755/2754
| 0.62851
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| 2.3.5.17.19.29 {{monzo|-1 -4 1 -1 1 1}}
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| 2784/2783
| 0.62196
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| 2.3.11.23.29 {{monzo|5 1 -2 -1 1}}
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| 3249/3248
| 0.53293
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| 2.3.7.19.29 {{monzo|-4 2 -1 2 -1}}
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| S57
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| 3451/3450
| 0.50173
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| 2.3.5.7.17.23.29 {{monzo|-1 -1 -2 1 1 -1 1}}
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| 3510/3509
| 0.49330
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| 2.3.5.11.13.29 {{monzo|1 3 1 -2 1 -1}}
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| 4641/4640
| 0.37307
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| 2.3.5.7.13.17.29 {{monzo|-5 1 -1 1 1 1 -1}}
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| 4785/4784
| 0.36184
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| 2.3.5.11.13.23.29 {{monzo|-4 1 1 1 -1 -1 1}}
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| 4901/4900
| 0.35328
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| 2.5.7.13.29 {{monzo|-2 -2 -2 2 1}}
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| 5104/5103
| 0.33922
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| 2.3.7.11.29 {{monzo|4 -6 -1 1 1}}
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| 5888/5887
| 0.29405
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| 2.7.23.29 {{monzo|8 -1 1 -2}}
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| 5916/5915
| 0.29266
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| 2.3.5.7.13.17.29 {{monzo|2 1 -1 -1 -2 1 1}}
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| 6670/6669
| 0.25957
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| 2.3.5.13.19.23.29 {{monzo|1 -3 1 -1 -1 1 1}}
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| 7106/7105
| 0.24365
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| 2.5.7.11.17.19.29 {{monzo|1 -1 -2 1 1 1 -1}}
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| 7425/7424
| 0.23318
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| 2.3.5.11.29 {{monzo|-8 3 2 1 -1}}
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| 7889/7888
| 0.21946
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| 2.7.17.23.29 {{monzo|-4 3 -1 1 -1}}
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| 8671/8670
| 0.19967
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| 2.3.5.13.17.23.29 {{monzo|-1 -1 -1 1 -2 1 1}}
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| 9802/9801
| 0.17663
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| 2.3.11.13.29 {{monzo|1 -4 -2 2 1}}
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| 10557/10556
| 0.16400
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| 2.3.7.13.17.23.29 {{monzo|-2 3 -1 -1 1 1 -1}}
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| 11340/11339
| 0.15267
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| 2.3.5.7.17.23.29 {{monzo|2 4 1 1 -1 -1 -1}}
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| 12006/12005
| 0.14420
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| 2.3.5.7.23.29 {{monzo|1 2 -1 -4 1 1}}
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| 12673/12672
| 0.13661
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| 2.3.11.19.23.29 {{monzo|-7 -2 -1 1 1 1}}
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| 13225/13224
| 0.13091
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| 2.3.5.19.23.29 {{monzo|-3 -1 2 -1 2 -1}}
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| S115
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| 13311/13310
| 0.13007
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| 2.3.5.11.17.29 {{monzo|-1 3 -1 -3 1 1}}
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| 13312/13311
| 0.13006
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| 2.3.13.17.29 {{monzo|10 -3 1 -1 -1}}
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| 13456/13455
| 0.12866
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| 2.3.5.13.23.29 {{monzo|4 -2 -1 -1 -1 2}}
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| S116
|-
| 19228/19227
| 0.090039
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| 2.3.11.13.17.19.23.29 {{monzo|2 -1 1 -1 -1 1 1 -1}}
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| 20736/20735
| 0.083491
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| 2.3.5.11.13.29 {{monzo|8 4 -1 -1 -1 -1}}
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| S144
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| 23751/23750
| 0.072893
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| 2.3.5.7.13.19.29 {{monzo|-1 2 -4 1 1 -1 1}}
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| 24795/24794
| 0.069823
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| 2.3.5.7.11.19.23.29 {{monzo|-1 2 1 -2 -1 1 -1 1}}
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| 25840/25839
| 0.067000
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| 2.3.5.11.17.19.29 {{monzo|4 -4 1 -1 1 1 -1}}
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| 27000/26999
| 0.064121
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| 2.3.5.7.19.29 {{monzo|3 3 3 -2 -1 -1}}
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| 30625/30624
| 0.056531
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| 2.3.5.7.11.20 {{monzo|-5 -1 4 2 -1 -1}}
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| S175
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| 30856/30855
| 0.056108
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| 2.3.5.7.11.17.19.29 {{monzo|3 -1 -1 1 -2 -1 1 1}}
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| 35322/35321
| 0.049014
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| 2.3.7.11.13.19.29 {{monzo|1 1 1 -1 -2 -1 2}}
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| 47125/47124
| 0.036737
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| 2.3.5.7.11.13.17.29 {{monzo|-2 -2 3 -1 -1 1 -1 1}}
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| 53361/53360
| 0.032444
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| 2.3.5.7.11.23.29 {{monzo|-4 2 -1 2 2 -1 -1}}
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| S231
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| 72501/72500
| 0.023879
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| 2.3.5.11.13.29 {{monzo|-2 1 -4 1 3 -1}}
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| 83521/83520
| 0.020728
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| 2.3.5.17.29 {{monzo|-6 -2 -1 4 -1}}
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| S289
|-
| 87465/87464
| 0.019794
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| 2.3.5.7.13.17.29 {{monzo|-3 1 1 3 -1 1 -2}}
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| 136851/136850
| 0.012651
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| 2.3.5.7.11.13.17.23.29 {{monzo|-1 1 -2 -1 2 1 -1 -1 1}}
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| 158950/158949
| 0.010892
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| 2.3.5.7.11.17.29 {{monzo|1 -3 2 -1 1 2 -2}}
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| 166635/166634
| 0.010389
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| 2.3.5.7.13.17.23.29 {{monzo|-1 2 1 1 -2 -1 2 -1}}
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| 168751/168750
| 0.010259
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| 2.3.5.11.23.29 {{monzo|-1 -3 -5 1 2 1}}
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| 176001/176000
| 0.0098365
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| 2.3.5.7.11.17.29 {{monzo|-7 1 -3 1 -1 2 1}}
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| 176176/176175
| 0.0098268
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| 2.3.5.7.11.13.29 {{monzo|4 -5 -2 1 2 1 -1}}
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| 184093/184092
| 0.0094042
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| 2.3.7.13.17.23.29 {{monzo|-2 -1 2 1 2 -2 -1}}
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| 240787/240786
| 0.0071899
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| 2.3.7.13.19.23.29 {{monzo|-1 -3 -3 -1 2 1 1}}
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| 244036/244035
| 0.0070942
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| 2.3.5.11.13.17.19.29 {{monzo|2 -2 -1 -1 2 -1 2 -1}}
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| S494
|-
| 303601/303600
| 0.0057023
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| 2.3.5.11.19.23.29 {{monzo|-4 -1 -2 -1 2 -1 2}}
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| S551
|-
| 410670/410669
| 0.0042156
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| 2.3.5.7.13.17.29 {{monzo|1 5 1 -2 2 -2 -1}}
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| 418761/418760
| 0.0041342
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| 2.3.5.7.17.19.23.29 {{monzo|-3 2 -1 1 2 -2 1 -1}}
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| 613089/613088
| 0.0028238
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| 2.3.7.17.23.29 {{monzo| -5 6 -2 -1 -1 2}}
| 2.3.7.17.23.29 {{monzo| -5 6 -2 -1 -1 2}}
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|
Line 2,094: Line 2,426:
| 2.3.5.7.13.19.29 {{monzo|-5 4 -3 -1 -1 4 -1}}
| 2.3.5.7.13.19.29 {{monzo|-5 4 -3 -1 -1 4 -1}}
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| S3249
|-
|-
| 18085705/18085704
| 18085705/18085704
Retrieved from "https://en.xen.wiki/w/List_of_superparticular_intervals"