[seqfan] Gauss-Kuzmin Frequency; Re: Signature sequences
Paul D Hanna
pauldhanna at juno.com
Thu May 29 06:35:06 CEST 2003
This subject is not exactly signature sequences, but is somewhat
related.
Below I append the union of the multiples m of the reciprocal
Gauss-Kuzmin frequencies of integers k, numbered in ascending order by n.
Here I denote Gauss-Kuzmin frequency of k by GK(k), where
GK(k) = log(1 + 1/(k*(k+2)) / log(2).
ANY PATTERNS TO THE SEQUENCES?
Notice one pattern of the values of n when m=1:
n=1, k=1, m/GK(k)= 2.409420839653463384
n=3, k=2, m/GK(k)= 5.884949192362146710
n=6, k=3, m/GK(k)=10.740053666281483132
n=11, k=4, m/GK(k)=16.865945877574243693
n=18, k=5, m/GK(k)=24.605097717735793855
n=24, k=6, m/GK(k)=33.616447248866527729
n=34, k=7, m/GK(k)=44.013936308554193692
n=45, k=8, m/GK(k)=55.797630484035824061
n=56, k=9, m/GK(k)=68.967563936505075182
n=68, k=10,m/GK(k)=83.523755900347857243
...
In each case, the occurrence of the next 'm=1' is predicted closely by
the value of floor(1/GK(k)). Is this due to the fact that
sum(k>0,GK(k))=1 ?
WHAT IS 'BEST-FIT' of GK?
Note that this construct causes the frequency of the values of {k} to
obey the Gauss-Kuzmin law for continued fractions.
My initial idea was to obtain the partial quotients {k} that would be the
'best-fit' of the Gauss-Kuzmin frequency, and this is what I proposed:
x= 0.5815803358828329856145...
= [0;1,1,2,1,1,3,2,1,1,1,4,2,1,3,1,2,1,5,1,1,2,1,3,6,1,4,...]
By construction, this constant would obey Khintchine's theorem.
But I wonder if the constant is truly the 'best-fit' to Gauss-Kuzmin, and
if such a constant could be defined.
See: http://www.math.niu.edu/~rusin/known-math/99/kusmin
At any rate, I find these sequences, {k} and {m}, interesting, as well as
the gaps between m/GK(k): which integers are not included in
{floor(m/GK(k))}?
Sequence {k}:
1,1,2,1,1,3,2,1,1,1,4,2,1,3,1,2,1,5,1,1,2,
1,3,6,1,4,2,1,1,1,2,3,1,7,1,2,1,5,1,4,2,1,
3,1,8,1,2,1,1,3,2,1,6,1,4,9,1,2,1,5,1,3,2,
1,1,1,2,10,1,4,3,1,7,2,1,1,1,2,1,3,5,1,11,
2,6,1,4,1,2,1,3,1,1,8,2,1,1,12,2,1,3,4,1,
1,5,2,1,1,3,2,1,7,1,6,1,2,13,4,1,9,3,1,2,
1,1,1,2,5,1,3,1,4,2,1,14,1,2,1,3,1,1,2,1,
10,8,6,1,4,2,1,3,5,1,1,7,2,15,1,1,2,3,1,1,
4,1,2,1,1,3,2,1,5,1,11,16,1,2,6,1,4,3,1,2,
9,1,1,2,1,1,3,1,2,1,7,4,5,1,8,2,1,17,3,1,
1,2,1,12,1,6,2,1,3,4,1,1,2,1,1,5,3,2,1,18,
10,1,1,2,4,1,3,1,2,1,1,7,2,1,1,3,6,1,5,2,
13,4,1,1,9,2,19,1,8,3,1,1,2,1,1,2,4,1,3,1,
1,2,5,1,11,1,2,3,1,6,1,20,4,1,2,7,1,1,14,3,
2,1,1,2,1,5,1,3,4,1,2,1,1,2,1,1,3,10,8,1,
Sequence {m}:
1,2,1,3,4,1,2,5,6,7,1,3,8,2,9,4,10,1,11,12,5,13,
3,1,14,2,6,15,16,17,7,4,18,1,19,8,20,2,21,3,9,22,5,
23,1,24,10,25,26,6,11,27,2,28,4,1,29,12,30,3,31,7,
13,32,33,34,14,1,35,5,8,36,2,15,37,38,39,16,40,9,4,
41,1,17,3,42,6,43,18,44,10,45,46,2,19,47,48,1,20,49,
11,7,50,51,5,21,52,53,12,22,54,3,55,4,56,23,1,8,57,
2,13,58,24,59,60,61,25,6,62,14,63,9,26,64,1,65,27,66,
15,67,68,28,69,2,3,5,70,10,29,71,16,7,72,73,4,30,1,
74,75,31,17,76,77,11,78,32,79,80,18,33,81,8,82,2,1,
83,34,6,84,12,19,85,35,3,86,87,36,88,89,20,90,37,91,
5,13,9,92,4,38,93,1,21,94,95,39,96,2,97,7,40,98,22,
14,99,100,41,101,102,10,23,42,103,1,3,104,105,43,15,
106,24,107,44,108,109,6,45,110,111,25,8,112,11,46,2,16,
113,114,4,47,1,115,5,26,116,117,48,118,119,49,17,120,
27,121,122,50,12,123,3,124,51,28,125,9,126,1,18,127,52,
7,128,129,2,29,53,130,131,54,132,13,133,30,19,134,55,
Sequence {floor(m/GK(k))}:
2,4,5,7,9,10,11,12,14,16,16,17,19,21,21,23,24,24,26,
28,29,31,32,33,33,33,35,36,38,40,41,42,43,44,45,47,48,
49,50,50,52,53,53,55,55,57,58,60,62,64,64,65,67,67,67,
68,69,70,72,73,74,75,76,77,79,81,82,83,84,84,85,86,88,
88,89,91,93,94,96,96,98,98,99,100,100,101,101,103,105,
106,107,108,110,111,111,113,115,116,117,118,118,118,120,
122,123,123,125,127,128,129,130,132,132,134,134,135,135,
135,137,137,139,139,141,142,144,146,147,147,149,150,151,
152,153,154,155,156,158,159,161,161,163,164,166,167,167,
168,168,169,170,171,171,172,173,175,176,176,177,178,180,
182,182,183,185,186,187,188,190,192,193,194,195,196,197,
198,199,199,200,201,202,203,204,204,205,206,207,209,211,
212,214,214,216,217,219,220,220,221,221,223,223,224,224,
225,226,228,229,231,233,233,235,235,236,236,237,238,240,
241,243,245,246,247,247,248,249,250,250,252,253,254,255,
Missing integers in {floor(m/GK(k))}:
1,3,6,8,13,15,18,20,22,25,27,30,34,37,39,46,51,54,56,59,
61,63,66,71,78,80,87,90,92,95,97,102,104,109,112,114,119,
121,124,126,131,133,136,138,140,143,145,148,157,160,162,
165,174,179,181,184,189,191,208,210,213,215,218,222,...
Any comments would be welcomed.
Thanks,
Paul
------------------------------------------------
. n k m m / GK(k)
. 1 1: 1. 2.409420839653463384
. 2 1: 2. 4.818841679306926769
. 3 2: 1. 5.884949192362146710
. 4 1: 3. 7.228262518960390154
. 5 1: 4. 9.637683358613853539
. 6 3: 1. 10.740053666281483132
. 7 2: 2. 11.769898384724293421
. 8 1: 5. 12.047104198267316923
. 9 1: 6. 14.456525037920780308
. 10 1: 7. 16.865945877574243693
. 11 4: 1. 16.979748018338516689
. 12 2: 3. 17.654847577086440132
. 13 1: 8. 19.275366717227707078
. 14 3: 2. 21.480107332562966265
. 15 1: 9. 21.684787556881170462
. 16 2: 4. 23.539796769448586843
. 17 1: 10. 24.094208396534633847
. 18 5: 1. 24.605097717735793855
. 19 1: 11. 26.503629236188097232
. 20 1: 12. 28.913050075841560617
. 21 2: 5. 29.424745961810733554
. 22 1: 13. 31.322470915495024002
. 23 3: 3. 32.220160998844449398
. 24 6: 1. 33.616447248866527729
. 25 1: 14. 33.731891755148487386
. 26 4: 2. 33.959496036677033379
. 27 2: 6. 35.309695154172880265
. 28 1: 15. 36.141312594801950771
. 29 1: 16. 38.550733434455414156
. 30 1: 17. 40.960154274108877541
. 31 2: 7. 41.194644346535026976
. 32 3: 4. 42.960214665125932530
. 33 1: 18. 43.369575113762340925
. 34 7: 1. 44.013936308554193692
. 35 1: 19. 45.778995953415804310
. 36 2: 8. 47.079593538897173687
. 37 1: 20. 48.188416793069267695
. 38 5: 2. 49.210195435471587710
. 39 1: 21. 50.597837632722731080
. 40 4: 3. 50.939244055015550069
. 41 2: 9. 52.964542731259320398
. 42 1: 22. 53.007258472376194465
. 43 3: 5. 53.700268331407415663
. 44 1: 23. 55.416679312029657849
. 45 8: 1. 55.797630484035824061
. 46 1: 24. 57.826100151683121234
. 47 2: 10. 58.849491923621467109
. 48 1: 25. 60.235520991336584619
. 49 1: 26. 62.644941830990048004
. 50 3: 6. 64.440321997688898796
. 51 2: 11. 64.734441115983613820
. 52 1: 27. 65.054362670643511388
. 53 6: 2. 67.232894497733055458
. 54 1: 28. 67.463783510296974773
. 55 4: 4. 67.918992073354066759
. 56 9: 1. 68.967563936505075182
. 57 1: 29. 69.873204349950438158
. 58 2: 12. 70.619390308345760531
. 59 1: 30. 72.282625189603901543
. 60 5: 3. 73.815293153207381565
. 61 1: 31. 74.692046029257364927
. 62 3: 7. 75.180375663970381929
. 63 2: 13. 76.504339500707907242
. 64 1: 32. 77.101466868910828312
. 65 1: 33. 79.510887708564291697
. 66 1: 34. 81.920308548217755082
. 67 2: 14. 82.389288693070053953
. 68 10: 1. 83.523755900347857243
. 69 1: 35. 84.329729387871218467
. 70 4: 5. 84.898740091692583449
. 71 3: 8. 85.920429330251865061
. 72 1: 36. 86.739150227524681851
. 73 7: 2. 88.027872617108387385
. 74 2: 15. 88.274237885432200664
. 75 1: 37. 89.148571067178145236
. 76 1: 38. 91.557991906831608621
. 77 1: 39. 93.967412746485072006
. 78 2: 16. 94.159187077794347375
. 79 1: 40. 96.376833586138535390
. 80 3: 9. 96.660482996533348194
. 81 5: 4. 98.420390870943175420
. 82 1: 41. 98.786254425791998775
. 83 11: 1. 99.466217884583035803
. 84 2: 17. 100.044136270156494085
. 85 6: 3. 100.849341746599583188
. 86 1: 42. 101.195675265445462160
. 87 4: 6. 101.878488110031100139
. 88 1: 43. 103.605096105098925545
. 89 2: 18. 105.929085462518640796
. 90 1: 44. 106.014516944752388930
. 91 3: 10. 107.400536662814831327
. 92 1: 45. 108.423937784405852314
. 93 1: 46. 110.833358624059315699
. 94 8: 2. 111.595260968071648122
. 95 2: 19. 111.814034654880787507
. 96 1: 47. 113.242779463712779084
. 97 1: 48. 115.652200303366242469
. 98 12: 1. 116.794957120786374235
. 99 2: 20. 117.698983847242934218
. 100 1: 49. 118.061621143019705853
. 101 3: 11. 118.140590329096314459
. 102 4: 7. 118.858236128369616829
. 103 1: 50. 120.471041982673169238
. 104 1: 51. 122.880462822326632623
. 105 5: 5. 123.025488588678969275
. 106 2: 21. 123.583933039605080929
. 107 1: 52. 125.289883661980096008
. 108 1: 53. 127.699304501633559393
. 109 3: 12. 128.880643995377797592
. 110 2: 22. 129.468882231967227640
. 111 1: 54. 130.108725341287022777
. 112 7: 3. 132.041808925662581078
. 113 1: 55. 132.518146180940486162
. 114 6: 4. 134.465788995466110917
. 115 1: 56. 134.927567020593949547. 116 2: 23.
135.353831424329374351
. 117 13: 1. 135.509978339807338529
. 118 4: 8. 135.837984146708133519
. 119 1: 57. 137.336987860247412932
. 120 9: 2. 137.935127873010150364
. 121 3: 13. 139.620697661659280725
. 122 1: 58. 139.746408699900876316
. 123 2: 24. 141.238780616691521062
. 124 1: 59. 142.155829539554339701
. 125 1: 60. 144.565250379207803086
. 126 1: 61. 146.974671218861266471
. 127 2: 25. 147.123729809053667773
. 128 5: 6. 147.630586306414763130
. 129 1: 62. 149.384092058514729855
. 130 3: 14. 150.360751327940763858
. 131 1: 63. 151.793512898168193240
. 132 4: 9. 152.817732165046650209
. 133 2: 26. 153.008679001415814484
. 134 1: 64. 154.202933737821656625
. 135 14: 1. 155.611284742426132435
. 136 1: 65. 156.612354577475120010
. 137 2: 27. 158.893628193777961195
. 138 1: 66. 159.021775417128583395
. 139 3: 15. 161.100804994222246990
. 140 1: 67. 161.431196256782046779
. 141 1: 68. 163.840617096435510164
. 142 2: 28. 164.778577386140107906
. 143 1: 69. 166.250037936088973549
. 144 10: 2. 167.047511800695714487
. 145 8: 3. 167.392891452107472183
. 146 6: 5. 168.082236244332638646
. 147 1: 70. 168.659458775742436934
. 148 4: 10. 169.797480183385166899
. 149 2: 29. 170.663526578502254617
. 150 1: 71. 171.068879615395900318
. 151 3: 16. 171.840858660503730123
. 152 5: 7. 172.235684024150556985
. 153 1: 72. 173.478300455049363703
. 154 1: 73. 175.887721294702827088
. 155 7: 4. 176.055745234216774771
. 156 2: 30. 176.548475770864401328
. 157 15: 1. 177.098878557435750322
. 158 1: 74. 178.297142134356290473
. 159 1: 75. 180.706562974009753858
. 160 2: 31. 182.433424963226548039
. 161 3: 17. 182.580912326785213256
. 162 1: 76. 183.115983813663217242
. 163 1: 77. 185.525404653316680627
. 164 4: 11. 186.777228201723683589
. 165 1: 78. 187.934825492970144012
. 166 2: 32. 188.318374155588694750
. 167 1: 79. 190.344246332623607397
. 168 1: 80. 192.753667172277070781
. 169 3: 18. 193.320965993066696389
. 170 2: 33. 194.203323347950841460
. 171 1: 81. 195.163088011930534166
. 172 5: 8. 196.840781741886350840
. 173 1: 82. 197.572508851583997551
. 174 11: 2. 198.932435769166071606
. 175 16: 1. 199.972761375538618998
. 176 1: 83. 199.981929691237460936
. 177 2: 34. 200.088272540312988171
. 178 6: 6. 201.698683493199166376
. 179 1: 84. 202.391350530890924321
. 180 4: 12. 203.756976220062200279
. 181 3: 19. 204.061019659348179521
. 182 1: 85. 204.800771370544387705
. 183 2: 35. 205.973221732675134882
. 184 9: 3. 206.902691809515225547
. 185 1: 86. 207.210192210197851090
. 186 1: 87. 209.619613049851314475
. 187 2: 36. 211.858170925037281593
. 188 1: 88. 212.029033889504777860
. 189 1: 89. 214.438454729158241244
. 190 3: 20. 214.801073325629662654
. 191 1: 90. 216.847875568811704629
. 192 2: 37. 217.743120117399428304
. 193 1: 91. 219.257296408465168014
. 194 7: 5. 220.069681542770968463
. 195 4: 13. 220.736724238400716968
. 196 5: 9. 221.445879459622144695
. 197 1: 92. 221.666717248118631399
. 198 8: 4. 223.190521936143296244
. 199 2: 38. 223.628069309761575015
. 200 1: 93. 224.076138087772094783
. 201 17: 1. 224.232934356877362998
. 202 3: 21. 225.541126991911145787
. 203 1: 94. 226.485558927425558168
. 204 1: 95. 228.894979767079021553
. 205 2: 39. 229.513018502123721726
. 206 1: 96. 231.304400606732484938
. 207 12: 2. 233.589914241572748470
. 208 1: 97. 233.713821446385948323
. 209 6: 7. 235.315130742065694105
. 210 2: 40. 235.397967694485868437
. 211 1: 98. 236.123242286039411707
. 212 3: 22. 236.281180658192628919
. 213 4: 14. 237.716472256739233658
. 214 1: 99. 238.532663125692875092
. 215 1: 100. 240.942083965346338477
. 216 2: 41. 241.282916886848015148
. 217 1: 101. 243.351504804999801862
. 218 1: 102. 245.760925644653265246
. 219 5: 10. 246.050977177357938550
. 220 3: 23. 247.021234324474112052
. 221 2: 42. 247.167866079210161859
. 222 1: 103. 248.170346484306728631
. 223 18: 1. 249.879398363598296293
. 224 10: 3. 250.571267701043571730
. 225 1: 104. 250.579767323960192016
. 226 1: 105. 252.989188163613655401
. 227 2: 43. 253.052815271572308570
. 228 4: 15. 254.696220275077750348
. 229 1: 106. 255.398609003267118786
. 230 3: 24. 257.761287990755595185
. 231 1: 107. 257.808029842920582170
. 232 2: 44. 258.937764463934455281
. 233 1: 108. 260.217450682574045555
. 234 1: 109. 262.626871522227508940
. 235 7: 6. 264.083617851325162156
. 236 2: 45. 264.822713656296601992
. 237 1: 110. 265.036292361880972325
. 238 1: 111. 267.445713201534435709
. 239 3: 25. 268.501341657037078318
. 240 6: 8. 268.931577990932221834
. 241 1: 112. 269.855134041187899094
. 242 5: 11. 270.656074895093732405
. 243 2: 46. 270.707662848658748703
. 244 13: 2. 271.019956679614677059
. 245 4: 16. 271.675968293416267038
. 246 1: 113. 272.264554880841362479
. 247 1: 114. 274.673975720494825864
. 248 9: 4. 275.870255746020300729
. 249 2: 47. 276.592612041020895414
. 250 19: 1. 276.912154047235477420
. 251 1: 115. 277.083396560148289248
. 252 8: 5. 278.988152420179120305
. 253 3: 26. 279.241395323318561450
. 254 1: 116. 279.492817399801752633
. 255 1: 117. 281.902238239455216018
. 256 2: 48. 282.477561233383042125
. 257 1: 118. 284.311659079108679403
. 258 1: 119. 286.721079918762142788
. 259 2: 49. 288.362510425745188835
. 260 4: 17. 288.655716311754783728
. 261 1: 120. 289.130500758415606172
. 262 3: 27. 289.981448989600044583
. 263 1: 121. 291.539921598069069557
. 264 1: 122. 293.949342437722532942
. 265 2: 50. 294.247459618107335546
. 266 5: 12. 295.261172612829526260
. 267 1: 123. 296.358763277375996327
. 268 11: 3. 298.398653653749107410
. 269 1: 124. 298.768184117029459711
. 270 2: 51. 300.132408810469482257
. 271 3: 28. 300.721502655881527716
. 272 1: 125. 301.177604956682923096
. 273 6: 9. 302.548025239798749564
. 274 1: 126. 303.587025796336386481
. 275 20: 1. 305.331201907743374651
. 276 4: 18. 305.635464330093300418
. 277 1: 127. 305.996446635989849866
. 278 2: 52. 306.017358002831628968
. 279 7: 7. 308.097554159879355849
. 280 1: 128. 308.405867475643313251
. 281 1: 129. 310.815288315296776635
. 282 14: 2. 311.222569484852264871
. 283 3: 29. 311.461556322163010849
. 284 2: 53. 311.902307195193775679
. 285 1: 130. 313.224709154950240020
. 286 1: 131. 315.634129994603703405
. 287 2: 54. 317.787256387555922390
. 288 1: 132. 318.043550834257166790
. 289 5: 13. 319.866270330565320115
. 290 1: 133. 320.452971673910630174
. 291 3: 30. 322.201609988444493981
. 292 4: 19. 322.615212348431817108
. 293 1: 134. 322.862392513564093559
. 294 2: 55. 323.672205579918069101
. 295 1: 135. 325.271813353217556944
. 296 1: 136. 327.681234192871020329
. 297 2: 56. 329.557154772280215812
. 298 1: 137. 330.090655032524483714
. 299 1: 138. 332.500075872177947098
. 300 3: 31. 332.941663654725977114
. 301 10: 4. 334.095023601391428974
. 302 8: 6. 334.785782904214944366
. 303 1: 139. 334.909496711831410483
. 304 21: 1. 335.136542333799851246
. 305 2: 57. 335.442103964642362523
. 306 6: 10. 336.164472488665277293
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On Wed, 28 May 2003 17:23:56 -0400 (EDT) "N. J. A. Sloane"
<njas at research.att.com> writes:
>
> Kerry Mitchell just posted a message to the math fun list
> which relates to sequences, and I will append it below.
>
> The OEIS prsently contains these examples:
>
> signature sequences (1): A007336 A007337 A023115 A023116 A023117
> A023118 A023119 A023120 A023121 A023122 A023123 A023124
> signature sequences (2): A023125 A023126 A023127 A023128 A023129
> A023130 A023131 A023132 A023133 A023134 A035796
>
> But the sig. seq. for phi seems to be missing - maybe someone
> could send it in.
>
> NJAS
>
>
> >Date: Wed, 28 May 2003 18:57:04 +0000
> >
> >Lately, I've been playing with the signature of irrational numbers.
> My
> >understanding, from what I've read on the web, is this: For a
> positive
> >irrational number x, form the numbers y = i + j*x, where i and j
> are both
> >positive integers. Since x is irrational, no 2 y values will be
> the same for
> >different i and j. Arrange the ys by size, then the sequence of i
> values is a
> >fractal sequence, and the signature of x.
> >
> >For x = phi ~ 1.618034, the first few entries are:
> >i j y
> >1 1 2.618033989
> >2 1 3.618033989
> >1 2 4.236067977
> >3 1 4.618033989
> >2 2 5.236067977
> >4 1 5.618033989
> >1 3 5.854101966
> >3 2 6.236067977
> >5 1 6.618033989
> >
> >And the signature begins: 1, 2, 1, 3, 2, 4, 1, 3, 5. If you strike
> the first
> >occurence of every integer in the sequence, you get the original
> sequence back,
> >which makes this a fractal sequence.
> >
> >My questions are:
> >
> >- What about the j sequence? From what I've seen experimentally,
> it seems to
> >be a fractal sequence, too. Why is the signature the i sequence as
> opposed to
> >the j sequence? What's known about the relation of the j sequence
> to the i
> >sequence?
> >
> >- For a limited set of integers (both i and j run from 1 to 50), I
> plotted i
> >vs. j, and the result was very interesting (I thought). You can
> find the
> >picture here:
> >
> w.fractalus.com/kerry/sigofphi.html
> >
> >The plot is one continuous zig-zag line which seems to never cross
> itself.
> >But, the angle of the line changes slightly, causing some areas to
> bunch up and
> >appear darker, and others to spread out and appear lighter. The
> overall effect
> >is of a series of rectangles drawn in different shades of gray.
> Can anyone
> >point me to other work that has been done on this?
> >
> >Thanks,
> >Kerry Mitchell
> >--
> >lkmitch at att.net
> >www.fractalus.com/kerry
> >
>
>
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