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→{{nowrap|Sk/S(k + 3)}} (three-particulars): add table of all 43-limit three-particulars up to S286/S289
Line 2,184: Line 2,184:


=== Table of three-particulars ===
=== Table of three-particulars ===
W.I.P.
{| class="wikitable center-all
{| class="wikitable center-all
|-
|-
Line 2,190: Line 2,189:
! Square relation
! Square relation
! Ratio
! Ratio
! Subgroup
|-
|-
| S2<sup>2</sup>⋅S3 = ([[3/2]])⋅([[4/3]])
| S2/S5
| ([[2/1]])<sup>2</sup>/([[2/1]])
| ( ([[4/1]])/([[5/2]]) )/( ([[5/2]])/([[2/1]]) )
| [[2/1]]
| [[32/25]]
| 2.5
|-
| S3/S6
| ( ([[5/2]])/([[2/1]]) )/( ([[2/1]])/([[7/4]]) )
| [[35/32]]
| 2.5.7
|-
| S4/S7
| ( ([[2/1]])/([[7/4]]) )/( ([[7/4]])/([[8/5]]) )
| [[256/245]]
| 2.5.7
|-
| S5/S8
| ( ([[7/4]])/([[8/5]]) )/( ([[8/5]])/([[3/2]]) )
| [[525/512]]
| 2.3.5.7
|-
| S6/S9
| ( ([[8/5]])/([[3/2]]) )/( ([[3/2]])/([[10/7]]) )
| [[64/63]]
| 2.3.7
|-
| S7/S10
| ( ([[3/2]])/([[10/7]]) )/( ([[10/7]])/([[11/8]]) )
| [[1617/1600]]
| 2.3.5.7.11
|-
| S8/S11
| ( ([[10/7]])/([[11/8]]) )/( ([[11/8]])/([[4/3]]) )
| [[2560/2541]]
| 2.3.5.7.11
|-
| S9/S12
| ( ([[11/8]])/([[4/3]]) )/( ([[4/3]])/([[13/10]]) )
| [[1287/1280]]
| 2.3.5.11.13
|-
| S10/S13
| ( ([[4/3]])/([[13/10]]) )/( ([[13/10]])/([[14/11]]) )
| [[5600/5577]]
| 2.3.5.7.11.13
|-
| S11/S14
| ( ([[13/10]])/([[14/11]]) )/( ([[14/11]])/([[5/4]]) )
| [[1573/1568]]
| 2.7.11.13
|-
| S12/S15
| ( ([[14/11]])/([[5/4]]) )/( ([[5/4]])/([[16/13]]) )
| [[3584/3575]]
| 2.5.7.11.13
|-
| S13/S16
| ( ([[5/4]])/([[16/13]]) )/( ([[16/13]])/([[17/14]]) )
| [[14365/14336]]
| 2.5.7.13.17
|-
| S14/S17
| ( ([[16/13]])/([[17/14]]) )/( ([[17/14]])/([[6/5]]) )
| [[18816/18785]]
| 2.3.5.7.13.17
|-
| S15/S18
| ( ([[17/14]])/([[6/5]]) )/( ([[6/5]])/([[19/16]]) )
| [[8075/8064]]
| 2.3.5.7.17.19
|-
| S16/S19
| ( ([[6/5]])/([[19/16]]) )/( ([[19/16]])/([[20/17]]) )
| [[6144/6137]]
| 2.3.17.19
|-
| S17/S20
| ( ([[19/16]])/([[20/17]]) )/( ([[20/17]])/([[7/6]]) )
| [[38437/38400]]
| 2.3.5.7.17.19
|-
| S18/S21
| ( ([[20/17]])/([[7/6]]) )/( ([[7/6]])/([[22/19]]) )
| [[15840/15827]]
| 2.3.5.7.11.17.19
|-
| S19/S22
| ( ([[7/6]])/([[22/19]]) )/( ([[22/19]])/([[23/20]]) )
| [[58121/58080]]
| 2.3.5.7.11.19.23
|-
| S20/S23
| ( ([[22/19]])/([[23/20]]) )/( ([[23/20]])/([[8/7]]) )
| [[70400/70357]]
| 2.5.7.11.19.23
|-
| S21/S24
| ( ([[23/20]])/([[8/7]]) )/( ([[8/7]])/([[25/22]]) )
| [[5635/5632]]
| 2.5.7.11.23
|-
| S22/S25
| ( ([[8/7]])/([[25/22]]) )/( ([[25/22]])/([[26/23]]) )
| [[100672/100625]]
| 2.5.7.11.13.23
|-
| S23/S26
| ( ([[25/22]])/([[26/23]]) )/( ([[26/23]])/([[9/8]]) )
| [[119025/118976]]
| 2.3.5.11.13.23
|-
| S24/S27
| ( ([[26/23]])/([[9/8]]) )/( ([[9/8]])/([[28/25]]) )
| [[46592/46575]]
| 2.3.5.7.13.23
|-
| S25/S28
| ( ([[9/8]])/([[28/25]]) )/( ([[28/25]])/([[29/26]]) )
| [[163125/163072]]
| 2.3.5.7.13.29
|-
| S26/S29
| ( ([[28/25]])/([[29/26]]) )/( ([[29/26]])/([[10/9]]) )
| [[37856/37845]]
| 2.3.5.7.13.29
|-
| S27/S30
| ( ([[29/26]])/([[10/9]]) )/( ([[10/9]])/([[31/28]]) )
| [[72819/72800]]
| 2.3.5.7.13.29.31
|-
| S28/S31
| ( ([[10/9]])/([[31/28]]) )/( ([[31/28]])/([[32/29]]) )
| [[250880/250821]]
| 2.3.5.7.29.31
|-
| S29/S32
| ( ([[31/28]])/([[32/29]]) )/( ([[32/29]])/([[11/10]]) )
| [[286781/286720]]
| 2.5.7.11.29.31
|-
| S30/S33
| ( ([[32/29]])/([[11/10]]) )/( ([[11/10]])/([[34/31]]) )
| [[108800/108779]]
| 2.5.11.17.29.31
|-
| S31/S34
| ( ([[11/10]])/([[34/31]]) )/( ([[34/31]])/([[35/32]]) )
| [[73997/73984]]
| 2.7.11.17.31
|-
| S32/S35
| ( ([[34/31]])/([[35/32]]) )/( ([[35/32]])/([[12/11]]) )
| [[417792/417725]]
| 2.3.5.7.11.17.31
|-
| S33/S36
| ( ([[35/32]])/([[12/11]]) )/( ([[12/11]])/([[37/34]]) )
| [[156695/156672]]
| 2.3.5.7.11.17.37
|-
| S34/S37
| ( ([[12/11]])/([[37/34]]) )/( ([[37/34]])/([[38/35]]) )
| [[527136/527065]]
| 2.3.5.7.11.17.19.37
|-
| S35/S38
| ( ([[37/34]])/([[38/35]]) )/( ([[38/35]])/([[13/12]]) )
| [[589225/589152]]
| 2.3.5.7.13.17.19.37
|-
| S36/S39
| ( ([[38/35]])/([[13/12]]) )/( ([[13/12]])/([[40/37]]) )
| [[43776/43771]]
| 2.3.7.13.19.37
|-
| S37/S40
| ( ([[13/12]])/([[40/37]]) )/( ([[40/37]])/([[41/38]]) )
| [[729677/729600]]
| 2.3.5.13.19.37.41
|-
| S38/S41
| ( ([[40/37]])/([[41/38]]) )/( ([[41/38]])/([[14/13]]) )
| [[808640/808561]]
| 2.5.7.13.19.37.41
|-
| S39/S42
| ( ([[41/38]])/([[14/13]]) )/( ([[14/13]])/([[43/40]]) )
| [[297947/297920]]
| 2.5.7.13.19.41.43
|-
| S40/S43
| ( ([[14/13]])/([[43/40]]) )/( ([[43/40]])/([[44/41]]) )
| [[985600/985517]]
| 2.5.7.11.13.41.43
|-
| S41/S44
| ( ([[43/40]])/([[44/41]]) )/( ([[44/41]])/([[15/14]]) )
| [[216849/216832]]
| 2.3.7.11.41.43
|-
| S42/S45
| ( ([[44/41]])/([[15/14]]) )/( ([[15/14]])/([[46/43]]) )
| [[396704/396675]]
| 2.3.5.7.11.23.41.43
>>> table([r for r in range(2,301) if len(fact( div_iv(S(r),S(r+3)) ))<=len(lim(43))])
|-
| S2/S5
| ( ([[4/1]])/([[5/2]]) )/( ([[5/2]])/([[2/1]]) )
| [[32/25]]
| 2.5
|-
| S3/S6
| ( ([[5/2]])/([[2/1]]) )/( ([[2/1]])/([[7/4]]) )
| [[35/32]]
| 2.5.7
|-
| S4/S7
| ( ([[2/1]])/([[7/4]]) )/( ([[7/4]])/([[8/5]]) )
| [[256/245]]
| 2.5.7
|-
| S5/S8
| ( ([[7/4]])/([[8/5]]) )/( ([[8/5]])/([[3/2]]) )
| [[525/512]]
| 2.3.5.7
|-
| S6/S9
| ( ([[8/5]])/([[3/2]]) )/( ([[3/2]])/([[10/7]]) )
| [[64/63]]
| 2.3.7
|-
| S7/S10
| ( ([[3/2]])/([[10/7]]) )/( ([[10/7]])/([[11/8]]) )
| [[1617/1600]]
| 2.3.5.7.11
|-
| S8/S11
| ( ([[10/7]])/([[11/8]]) )/( ([[11/8]])/([[4/3]]) )
| [[2560/2541]]
| 2.3.5.7.11
|-
| S9/S12
| ( ([[11/8]])/([[4/3]]) )/( ([[4/3]])/([[13/10]]) )
| [[1287/1280]]
| 2.3.5.11.13
|-
| S10/S13
| ( ([[4/3]])/([[13/10]]) )/( ([[13/10]])/([[14/11]]) )
| [[5600/5577]]
| 2.3.5.7.11.13
|-
| S11/S14
| ( ([[13/10]])/([[14/11]]) )/( ([[14/11]])/([[5/4]]) )
| [[1573/1568]]
| 2.7.11.13
|-
| S12/S15
| ( ([[14/11]])/([[5/4]]) )/( ([[5/4]])/([[16/13]]) )
| [[3584/3575]]
| 2.5.7.11.13
|-
| S13/S16
| ( ([[5/4]])/([[16/13]]) )/( ([[16/13]])/([[17/14]]) )
| [[14365/14336]]
| 2.5.7.13.17
|-
| S14/S17
| ( ([[16/13]])/([[17/14]]) )/( ([[17/14]])/([[6/5]]) )
| [[18816/18785]]
| 2.3.5.7.13.17
|-
| S15/S18
| ( ([[17/14]])/([[6/5]]) )/( ([[6/5]])/([[19/16]]) )
| [[8075/8064]]
| 2.3.5.7.17.19
|-
| S16/S19
| ( ([[6/5]])/([[19/16]]) )/( ([[19/16]])/([[20/17]]) )
| [[6144/6137]]
| 2.3.17.19
|-
| S17/S20
| ( ([[19/16]])/([[20/17]]) )/( ([[20/17]])/([[7/6]]) )
| [[38437/38400]]
| 2.3.5.7.17.19
|-
| S18/S21
| ( ([[20/17]])/([[7/6]]) )/( ([[7/6]])/([[22/19]]) )
| [[15840/15827]]
| 2.3.5.7.11.17.19
|-
| S19/S22
| ( ([[7/6]])/([[22/19]]) )/( ([[22/19]])/([[23/20]]) )
| [[58121/58080]]
| 2.3.5.7.11.19.23
|-
| S20/S23
| ( ([[22/19]])/([[23/20]]) )/( ([[23/20]])/([[8/7]]) )
| [[70400/70357]]
| 2.5.7.11.19.23
|-
| S21/S24
| ( ([[23/20]])/([[8/7]]) )/( ([[8/7]])/([[25/22]]) )
| [[5635/5632]]
| 2.5.7.11.23
|-
| S22/S25
| ( ([[8/7]])/([[25/22]]) )/( ([[25/22]])/([[26/23]]) )
| [[100672/100625]]
| 2.5.7.11.13.23
|-
| S23/S26
| ( ([[25/22]])/([[26/23]]) )/( ([[26/23]])/([[9/8]]) )
| [[119025/118976]]
| 2.3.5.11.13.23
|-
| S24/S27
| ( ([[26/23]])/([[9/8]]) )/( ([[9/8]])/([[28/25]]) )
| [[46592/46575]]
| 2.3.5.7.13.23
|-
| S25/S28
| ( ([[9/8]])/([[28/25]]) )/( ([[28/25]])/([[29/26]]) )
| [[163125/163072]]
| 2.3.5.7.13.29
|-
| S26/S29
| ( ([[28/25]])/([[29/26]]) )/( ([[29/26]])/([[10/9]]) )
| [[37856/37845]]
| 2.3.5.7.13.29
|-
| S27/S30
| ( ([[29/26]])/([[10/9]]) )/( ([[10/9]])/([[31/28]]) )
| [[72819/72800]]
| 2.3.5.7.13.29.31
|-
| S28/S31
| ( ([[10/9]])/([[31/28]]) )/( ([[31/28]])/([[32/29]]) )
| [[250880/250821]]
| 2.3.5.7.29.31
|-
| S29/S32
| ( ([[31/28]])/([[32/29]]) )/( ([[32/29]])/([[11/10]]) )
| [[286781/286720]]
| 2.5.7.11.29.31
|-
| S30/S33
| ( ([[32/29]])/([[11/10]]) )/( ([[11/10]])/([[34/31]]) )
| [[108800/108779]]
| 2.5.11.17.29.31
|-
| S31/S34
| ( ([[11/10]])/([[34/31]]) )/( ([[34/31]])/([[35/32]]) )
| [[73997/73984]]
| 2.7.11.17.31
|-
| S32/S35
| ( ([[34/31]])/([[35/32]]) )/( ([[35/32]])/([[12/11]]) )
| [[417792/417725]]
| 2.3.5.7.11.17.31
|-
| S33/S36
| ( ([[35/32]])/([[12/11]]) )/( ([[12/11]])/([[37/34]]) )
| [[156695/156672]]
| 2.3.5.7.11.17.37
|-
| S34/S37
| ( ([[12/11]])/([[37/34]]) )/( ([[37/34]])/([[38/35]]) )
| [[527136/527065]]
| 2.3.5.7.11.17.19.37
|-
| S35/S38
| ( ([[37/34]])/([[38/35]]) )/( ([[38/35]])/([[13/12]]) )
| [[589225/589152]]
| 2.3.5.7.13.17.19.37
|-
| S36/S39
| ( ([[38/35]])/([[13/12]]) )/( ([[13/12]])/([[40/37]]) )
| [[43776/43771]]
| 2.3.7.13.19.37
|-
| S37/S40
| ( ([[13/12]])/([[40/37]]) )/( ([[40/37]])/([[41/38]]) )
| [[729677/729600]]
| 2.3.5.13.19.37.41
|-
| S38/S41
| ( ([[40/37]])/([[41/38]]) )/( ([[41/38]])/([[14/13]]) )
| [[808640/808561]]
| 2.5.7.13.19.37.41
|-
| S39/S42
| ( ([[41/38]])/([[14/13]]) )/( ([[14/13]])/([[43/40]]) )
| [[297947/297920]]
| 2.5.7.13.19.41.43
|-
| S40/S43
| ( ([[14/13]])/([[43/40]]) )/( ([[43/40]])/([[44/41]]) )
| [[985600/985517]]
| 2.5.7.11.13.41.43
|-
| S41/S44
| ( ([[43/40]])/([[44/41]]) )/( ([[44/41]])/([[15/14]]) )
| [[216849/216832]]
| 2.3.7.11.41.43
|-
| S42/S45
| ( ([[44/41]])/([[15/14]]) )/( ([[15/14]])/([[46/43]]) )
| [[396704/396675]]
| 2.3.5.7.11.23.41.43
|-
| S286/S289
| ( ([[96/95]])/([[289/286]]) )/( ([[289/286]])/([[290/287]]) )
| [[455440128/455440013]]
| 2.3.7.11.13.17.19.29.41
|}
|}


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