## Poll

7 votes (41.17%) | |||

No votes (0%) | |||

2 votes (11.76%) | |||

2 votes (11.76%) | |||

2 votes (11.76%) | |||

6 votes (35.29%) | |||

No votes (0%) | |||

4 votes (23.52%) | |||

2 votes (11.76%) | |||

1 vote (5.88%) |

**17 members have voted**

Quote:gordonm888What are the results for a divisor of 13?

Damn Excel doesn't handle big numbers very well, so my sample size is rather small:

Divisor | Yes | No | Ratio | Expected |
---|---|---|---|---|

2 | 44 | 57 | 43.6% | 50.0% |

3 | 37 | 66 | 35.9% | 33.3% |

5 | 35 | 72 | 32.7% | 20.0% |

7 | 33 | 76 | 30.3% | 14.3% |

11 | 52 | 60 | 46.4% | 9.1% |

13 | 4 | 109 | 3.5% | 7.7% |

So 13's are underrepresented. I think it is indeed significant how often 11 pops up and how often 13 does not.

I think I'll add more code to check the factorization for larger numbers. I can handle numbers up to 2^64 in C++. Projects like this make me jealous of the people who have Mathematica.

Quote:I have taken all the primes from 2-1,000,000 and converted them into different bases/radices from 2-100+ and looked for frequency of integers in the digits (ignoring the first and last digit of each prime.) It is extraordinarily regular, with a digit frequency that has very little variance around the expected number. Given that as a background, I am surprised to see your large discrepancy in the frequency of divisor 11. It seems like it might be significant.

Can you show me a table to illustrate what you mean?

Prime | Count in 2 to 405 | Ratio | Expected |
---|---|---|---|

2 | 181 | 44.8% | 50.0% |

3 | 139 | 34.4% | 33.3% |

5 | 139 | 34.4% | 20.0% |

7 | 101 | 25.0% | 14.3% |

11 | 108 | 26.7% | 9.1% |

13 | 25 | 6.2% | 7.7% |

17 | 17 | 4.2% | 5.9% |

23 | 18 | 4.5% | 4.3% |

29 | 11 | 2.7% | 3.4% |

This shows a huge surplus of total partitions divisible by 5, 7, and 11, especially 11. You may recall these are the same primes in the Ramanujan identities. Very interesting...

- Semi-prime

- k-almost prime

- Sphenic number

https://en.wikipedia.org/wiki/Table_of_prime_factors?wprov=sfti1

Can you run your code for partitions of 10, 11, 12, 13, 14 and 15 and PM or email the lists to me please? I don't have the software to run the code myself.

Quote:WizardCan you show me a table to illustrate what you mean?

Sorry, I spent some time looking through my many spreadsheets and I didn't find the work that I had mentioned. I did find some spreadsheets with the primes from 3-19,997 converted into prime number radices (bases) so I used them to produce the (modest) results shown in the table below.

Explanation: This table is the frequency of digits (as shown in the first column) in prime numbers converted into base 2, 5, 7,and 11 - in which I only count the digits as they appear in the "2nd to last digit" and "3rd to last digit" of each prime number.

So in the 2nd row I am counting the frequency of digits 1-4 as they appear in all the primes from 29-19997, when those primes are written in Base 5 -and only counting the digits in the 5^3 and 5^2 columns of the prime numbers.

As an example the prime number 19,917 is written in base 5 as 1114202, so I count one "2" and one "0" as shown in bold in the number.

Digits | Base 3: 11-19997 | Base 5: 29-19997 | Base 7: 53-19997 | Base 11: 127-19997 |
---|---|---|---|---|

0 | 1492 | 885 | 661 | 410 |

1 | 1503 | 917 | 627 | 409 |

2 | 1517 | 916 | 633 | 394 |

3 | 898 | 634 | 412 | |

4 | 885 | 649 | 396 | |

5 | 641 | 404 | ||

6 | 645 | 414 | ||

7 | 417 | |||

8 | 423 | |||

9 | 388 | |||

10 | 393 |

This was just a quick effort to show the kind of work that I had done. The original work was on every prime less than 1,000,000 and the variance in the calculated frequencies of various digits seemed pretty damn small, as I remember it.

Quote:rsactuaryA favor of the Wiz:

Can you run your code for partitions of 10, 11, 12, 13, 14 and 15 and PM or email the lists to me please? I don't have the software to run the code myself.

Does this table not answer your question?

n | Partitions of n | Factorization of Partitions of n |
---|---|---|

2 | 2 | 2 |

3 | 3 | 3 |

4 | 5 | 5 |

5 | 7 | 7 |

6 | 11 | 11 |

7 | 15 | 3,5 |

8 | 22 | 2,11 |

9 | 30 | 2,3,5 |

10 | 42 | 2,3,7 |

11 | 56 | 2,2,2,7 |

12 | 77 | 7,11 |

13 | 101 | 101 |

14 | 135 | 3,3,3,5 |

15 | 176 | 2,2,2,2,11 |

16 | 231 | 3,7,11 |

17 | 297 | 3,3,3,11 |

18 | 385 | 5,7,11 |

19 | 490 | 2,5,7,7 |

20 | 627 | 3,11,19 |

21 | 792 | 2,2,2,3,3,11 |

22 | 1002 | 2,3,167 |

23 | 1255 | 5,251 |

24 | 1575 | 3,3,5,5,7 |

25 | 1958 | 2,11,89 |

26 | 2436 | 2,2,3,7,29 |

27 | 3010 | 2,5,7,43 |

28 | 3718 | 2,11,13,13 |

29 | 4565 | 5,11,83 |

30 | 5604 | 2,2,3,467 |

31 | 6842 | 2,11,311 |

32 | 8349 | 3,11,11,23 |

33 | 10143 | 3,3,7,7,23 |

34 | 12310 | 2,5,1231 |

35 | 14883 | 3,11,11,41 |

36 | 17977 | 17977 |

37 | 21637 | 7,11,281 |

38 | 26015 | 5,11,11,43 |

39 | 31185 | 3,3,3,3,5,7,11 |

40 | 37338 | 2,3,7,7,127 |

41 | 44583 | 3,7,11,193 |

42 | 53174 | 2,11,2417 |

43 | 63261 | 3,3,3,3,11,71 |

44 | 75175 | 5,5,31,97 |

45 | 89134 | 2,41,1087 |

46 | 105558 | 2,3,73,241 |

47 | 124754 | 2,7,7,19,67 |

48 | 147273 | 3,7,7013 |

49 | 173525 | 5,5,11,631 |

50 | 204226 | 2,11,9283 |

51 | 239943 | 3,11,11,661 |

52 | 281589 | 3,7,11,23,53 |

53 | 329931 | 3,3,7,5237 |

54 | 386155 | 5,7,11,17,59 |

55 | 451276 | 2,2,7,71,227 |

56 | 526823 | 11,47,1019 |

57 | 614154 | 2,3,102359 |

58 | 715220 | 2,2,5,11,3251 |

59 | 831820 | 2,2,5,11,19,199 |

60 | 966467 | 17,139,409 |

61 | 1121505 | 3,5,7,11,971 |

62 | 1300156 | 2,2,11,13,2273 |

63 | 1505499 | 3,113,4441 |

64 | 1741630 | 2,5,11,71,223 |

65 | 2012558 | 2,1006279 |

66 | 2323520 | 2,2,2,2,2,2,5,53,137 |

67 | 2679689 | 1181,2269 |

68 | 3087735 | 3,5,7,7,4201 |

69 | 3554345 | 5,641,1109 |

70 | 4087968 | 2,2,2,2,2,3,97,439 |

71 | 4697205 | 3,5,313147 |

72 | 5392783 | 11,139,3527 |

73 | 6185689 | 23,131,2053 |

74 | 7089500 | 2,2,5,5,5,11,1289 |

75 | 8118264 | 2,2,2,3,7,11,23,191 |

76 | 9289091 | 7,1327013 |

77 | 10619863 | 10619863 |

78 | 12132164 | 2,2,11,103,2677 |

79 | 13848650 | 2,5,5,173,1601 |

80 | 15796476 | 2,2,3,3,227,1933 |

81 | 18004327 | 11,1636757 |

82 | 20506255 | 5,7,7,7,11,1087 |

83 | 23338469 | 7,11,303097 |

84 | 26543660 | 2,2,5,11,13,9281 |

85 | 30167357 | 11,11,249317 |

86 | 34262962 | 2,23,37,41,491 |

87 | 38887673 | 11,3535243 |

88 | 44108109 | 3,3,83,137,431 |

89 | 49995925 | 5,5,7,7,40813 |

90 | 56634173 | 2473,22901 |

91 | 64112359 | 29,373,5927 |

92 | 72533807 | 3371,21517 |

93 | 82010177 | 59,1390003 |

94 | 92669720 | 2,2,2,5,11,13,17,953 |

95 | 104651419 | 283,369793 |

96 | 118114304 | 2,2,2,2,2,2,2,2,2,2,2,7,7,11,107 |

97 | 133230930 | 2,3,5,7,29,131,167 |

98 | 150198136 | 2,2,2,11,1706797 |

99 | 169229875 | 5,5,5,1353839 |

100 | 190569292 | 2,2,43,59,89,211 |

101 | 214481126 | 2,31,3459373 |

102 | 241265379 | 3,2423,33191 |

103 | 271248950 | 2,5,5,7,774997 |

104 | 304801365 | 3,5,11,1847281 |

105 | 342325709 | 11,43,43,16831 |

106 | 384276336 | 2,2,2,2,3,8005757 |

107 | 431149389 | 3,11,173,75521 |

108 | 483502844 | 2,2,11,10988701 |

109 | 541946240 | 2,2,2,2,2,2,2,5,11,23,3347 |

110 | 607163746 | 2,7,4049,10711 |

111 | 679903203 | 3,7,67,483229 |

112 | 761002156 | 2,2,190250539 |

113 | 851376628 | 2,2,212844157 |

114 | 952050665 | 5,193,986581 |

115 | 1064144451 | 3,61,67,229,379 |

116 | 1188908248 | 2,2,2,11,11,157,7823 |

117 | 1327710076 | 2,2,7,7,11,615821 |

118 | 1482074143 | 11,197,827,827 |

119 | 1653668665 | 5,11,30066703 |

120 | 1844349560 | 2,2,2,5,47,981037 |

121 | 2056148051 | 461,4460191 |

122 | 2291320912 | 2,2,2,2,9013,15889 |

123 | 2552338241 | 79,32308079 |

124 | 2841940500 | 2,2,3,5,5,5,7,31,8731 |

125 | 3163127352 | 2,2,2,3,7,11,59,67,433 |

126 | 3519222692 | 2,2,89,379,26083 |

127 | 3913864295 | 5,11,67,1062107 |

128 | 4351078600 | 2,2,2,5,5,11,17,317,367 |

129 | 4835271870 | 2,3,3,5,11,13,157,2393 |

130 | 5371315400 | 2,2,2,5,5,11,157,15551 |

131 | 5964539504 | 2,2,2,2,7,7,7,11,29,3407 |

132 | 6620830889 | 6620830889 |

133 | 7346629512 | 2,2,2,3,3,1319,77359 |

134 | 8149040695 | 5,17,89,1077203 |

135 | 9035836076 | 2,2,59,569,67289 |

136 | 10015581680 | 2,2,2,2,5,13,31,41,7577 |

137 | 11097645016 | 2,2,2,17,1367,59693 |

138 | 12292341831 | 3,3,7,7,11,733,3457 |

139 | 13610949895 | 5,79,34458101 |

140 | 15065878135 | 5,7,11,39132151 |

141 | 16670689208 | 2,2,2,29,31,991,2339 |

142 | 18440293320 | 2,2,2,3,3,5,127,403331 |

143 | 20390982757 | 7589,2686913 |

144 | 22540654445 | 5,5807,776327 |

145 | 24908858009 | 7,7,53,73,83,1583 |

146 | 27517052599 | 53197,517267 |

147 | 30388671978 | 2,3,7,24151,29959 |

148 | 33549419497 | 11,73,41780099 |

149 | 37027355200 | 2,2,2,2,2,2,5,5,11,11,11,17387 |

150 | 40853235313 | 11,17,197,1108967 |

151 | 45060624582 | 2,3,7510104097 |

152 | 49686288421 | 7,11,751,859223 |

153 | 54770336324 | 2,2,11,34513,36067 |

154 | 60356673280 | 2,2,2,2,2,2,2,2,5,37,1274423 |

155 | 66493182097 | 19,8087,432749 |

156 | 73232243759 | 463,1777,89009 |

157 | 80630964769 | 80630964769 |

158 | 88751778802 | 2,79,691,853,953 |

159 | 97662728555 | 5,7,29,67,1436111 |

160 | 107438159466 | 2,3,3,11,443,1224869 |

161 | 118159068427 | 797,148254791 |

162 | 129913904637 | 3,3,11,127,10332769 |

163 | 142798995930 | 2,3,5,4759966531 |

164 | 156919475295 | 3,3,5,11,14867,21323 |

165 | 172389800255 | 5,313,1543,71389 |

166 | 189334822579 | 7,7,37,53,1277,1543 |

167 | 207890420102 | 2,19,73,503,148991 |

168 | 228204732751 | 228204732751 |

169 | 250438925115 | 3,3,3,5,1855103149 |

170 | 274768617130 | 2,5,7,47,83516297 |

171 | 301384802048 | 2,2,2,2,2,2,2,2,11,11,1609,6047 |

172 | 330495499613 | 103,2351,1364821 |

173 | 362326859895 | 3,5,7,11,13,37,652189 |

174 | 397125074750 | 2,5,5,5,103,1627,9479 |

175 | 435157697830 | 2,5,43515769783 |

176 | 476715857290 | 2,5,443,107610803 |

177 | 522115831195 | 5,7,97,153789641 |

178 | 571701605655 | 3,5,17,2241967081 |

179 | 625846753120 | 2,2,2,2,2,5,277,3467,4073 |

180 | 684957390936 | 2,2,2,3,7,7,13,59,643,1181 |

181 | 749474411781 | 3,249824803927 |

182 | 819876908323 | 11,23,27967,115873 |

183 | 896684817527 | 61,293,50169799 |

184 | 980462880430 | 2,5,11,107,83301859 |

185 | 1071823774337 | 3469,6653,46441 |

186 | 1171432692373 | 1171432692373 |

187 | 1280011042268 | 2,2,7,7,18713,348991 |

188 | 1398341745571 | 1398341745571 |

189 | 1527273599625 | 3,3,3,3,5,5,5,61,127,19471 |

190 | 1667727404093 | 317,7283,722363 |

191 | 1820701100652 | 2,2,3,7,11171,1940293 |

192 | 1987276856363 | 23,503,171776027 |

193 | 2168627105469 | 3,11,71,269,3440807 |

194 | 2366022741845 | 5,7,7,521,18535961 |

195 | 2580840212973 | 3,11,78207279181 |

196 | 2814570987591 | 3,11,89,958314943 |

197 | 3068829878530 | 2,5,13,43,257,2136131 |

198 | 3345365983698 | 2,3,3,185853665761 |

199 | 3646072432125 | 3,3,5,5,5,3240953273 |

200 | 3972999029388 | 2,2,3,331083252449 |

201 | 4328363658647 | 7,19,23,47,4441,6779 |

202 | 4714566886083 | 3,837673,1876057 |

203 | 5134205287973 | 151,34001359523 |

204 | 5590088317495 | 5,11,19,5349366811 |

205 | 6085253859260 | 2,2,5,304262692963 |

206 | 6622987708040 | 2,2,2,5,11,15052244791 |

207 | 7206841706490 | 2,3,3,5,829,5399,17891 |

208 | 7840656226137 | 3,7,373364582197 |

209 | 8528581302375 | 3,5,5,5,19,197,2083,2917 |

210 | 9275102575355 | 5,487,1091,3491363 |

211 | 10085065885767 | 3,17,61,227,14280811 |

212 | 10963707205259 | 10963707205259 |

213 | 11916681236278 | 2,7,13,31,79,131,409,499 |

214 | 12950095925895 | 3,5,2351,7573,48491 |

215 | 14070545699287 | 7,7,11,11,23497,100999 |

216 | 15285151248481 | 15285151248481 |

217 | 16601598107914 | 2,2140069,3878753 |

218 | 18028182516671 | 53,340154387107 |

219 | 19573856161145 | 5,11,13,313,87463331 |

220 | 21248279009367 | 3,7082759669789 |

221 | 23061871173849 | 3,223,138283,249287 |

222 | 25025873760111 | 3,7,17,1259,55679497 |

223 | 27152408925615 | 3,5,23,353,4801,46439 |

224 | 29454549941750 | 2,5,5,5,16301,7227667 |

225 | 31946390696157 | 3,7,37,97,2887,146819 |

226 | 34643126322519 | 3,11,13,13,79,78630193 |

227 | 37561133582570 | 2,5,13,173,449,3719657 |

228 | 40718063627362 | 2,11,23,80470481477 |

229 | 44132934884255 | 5,7,7,191,943112189 |

230 | 47826239745920 | 2,2,2,2,2,2,2,5,109843,680321 |

231 | 51820051838712 | 2,2,2,3,59,59,620272573 |

232 | 56138148670947 | 3,7,32009,83515423 |

233 | 60806135438329 | 307,347,9749,58549 |

234 | 65851585970275 | 5,5,7,376294776973 |

235 | 71304185514919 | 7,10186312216417 |

236 | 77195892663512 | 2,2,2,7,7,263,748776797 |

237 | 83561103925871 | 11,11,13,17,677,4615703 |

238 | 90436839668817 | 3,7591,3971230829 |

239 | 97862933703585 | 3,5,1114697,5852887 |

240 | 105882246722733 | 3,13,13,83,2516153293 |

241 | 114540884553038 | 2,57270442276519 |

242 | 123888443077259 | 7,31,570914484227 |

243 | 133978259344888 | 2,2,2,7,7,97,5783,609289 |

244 | 144867692496445 | 5,14813,1955953453 |

245 | 156618412527946 | 2,4007,19543101139 |

246 | 169296722391554 | 2,431,18433,10654799 |

247 | 182973889854026 | 2,77899,1174430287 |

248 | 197726516681672 | 2,2,2,11,79,28441673861 |

249 | 213636919820625 | 3,3,3,5,5,5,5,7,277,6529121 |

250 | 230793554364681 | 3,7,11,10037,99542323 |

251 | 249291451168559 | 887,281050114057 |

252 | 269232701252579 | 4177,64455997427 |

253 | 290726957916112 | 2,2,2,2,53,73,4696416353 |

254 | 313891991306665 | 5,17,643,9649,595207 |

255 | 338854264248680 | 2,2,2,5,631,859,1429,10937 |

256 | 365749566870782 | 2,91381,2001234211 |

257 | 394723676655357 | 3,7,18796365555017 |

258 | 425933084409356 | 2,2,545863,195073253 |

259 | 459545750448675 | 3,5,5,11,234197,2378447 |

260 | 495741934760846 | 2,31,53,150864861461 |

261 | 534715062908609 | 11,41,349,20389,166619 |

262 | 576672674947168 | 2,2,2,2,2,19,439,2160534839 |

263 | 621837416509615 | 5,419,29867,9938051 |

264 | 670448123060170 | 2,5,7,7,13,41,1907,1346143 |

265 | 722760953690372 | 2,2,419,8167,52802941 |

266 | 779050629562167 | 3,3,3,28853727020821 |

267 | 839611730366814 | 2,3,53,6857,385050089 |

268 | 904760108316360 | 2,2,2,3,3,5,2513222523101 |

269 | 974834369944625 | 5,5,5,173,45079046009 |

270 | 1050197489931117 | 3,11,83,47309,8104667 |

271 | 1131238503938606 | 2,7,3041,26571111569 |

272 | 1218374349844333 | 11,317333,349038091 |

273 | 1312051800816215 | 5,499,2003,9829,26711 |

274 | 1412749565173450 | 2,5,5,7,7,173,4327,770311 |

275 | 1520980492851175 | 5,5,60839219714047 |

276 | 1637293969337171 | 167,1777,1931,2857199 |

277 | 1762278433057269 | 3,3,29,6752024647729 |

278 | 1896564103591584 | 2,2,2,2,2,3,7,7,7,23,2504230711 |

279 | 2040825852575075 | 5,5,373,218855319311 |

280 | 2195786311682516 | 2,2,7,389,3853,52321891 |

281 | 2362219145337711 | 3,11,37,983809,1966499 |

282 | 2540952590045698 | 2,29,59,619,1199570861 |

283 | 2732873183547535 | 5,7,7,53,210463857031 |

284 | 2938929793929555 | 3,5,790189,247951633 |

285 | 3160137867148997 | 7,7,7,52517,175433287 |

286 | 3397584011986773 | 3,3,3,7,43,547,764280217 |

287 | 3652430836071053 | 73,50033299124261 |

288 | 3925922161489422 | 2,3,103,2162249,2937971 |

289 | 4219388528587095 | 3,5,263,1069553492671 |

290 | 4534253126900886 | 2,3,227,9349,356093047 |

291 | 4872038056472084 | 2,2,1218009514118021 |

292 | 5234371069753672 | 2,2,2,7,7,11,97,223,56118901 |

293 | 5622992691950605 | 5,7087043,158683747 |

294 | 6039763882095515 | 5,11,13,73,1187,7027,13873 |

295 | 6486674127079088 | 2,2,2,2,461,1033,851335711 |

296 | 6965850144195831 | 3,7,83,5783,691072399 |

297 | 7479565078510584 | 2,2,2,3,3,3413,6971,4366289 |

298 | 8030248384943040 | 2,2,2,2,2,2,3,5,4801,1742312449 |

299 | 8620496275465025 | 5,5,7,17,2897645806879 |

300 | 9253082936723602 | 2,137,1021,33075784213 |

301 | 9930972392403501 | 3,6491,509986771037 |

302 | 10657331232548839 | 10657331232548839 |

303 | 11435542077822104 | 2,2,2,11,53,2451874373461 |

304 | 12269218019229465 | 3,5,7,257,617,1249,589993 |

305 | 13162217895057704 | 2,2,2,11,117101,1277279083 |

306 | 14118662665280005 | 5,7,23,67,79,2357,1405841 |

307 | 15142952738857194 | 2,3,3,41,607,33803799259 |

308 | 16239786535829663 | 11149,68473,21272819 |

309 | 17414180133147295 | 5,12650773,275306183 |

310 | 18671488299600364 | 2,2,181,503,51271070537 |

311 | 20017426762576945 | 5,13,31,199,33479,1491103 |

312 | 21458096037352891 | 7,15493,217027,911683 |

313 | 23000006655487337 | 7,7,7,41,1635497877799 |

314 | 24650106150830490 | 2,3,3,5,11,59,422018595289 |

315 | 26415807633566326 | 2,691,1609,11879544977 |

316 | 28305020340996003 | 3,3,11,113,233,10859101993 |

317 | 30326181989842964 | 2,2,557,44111,308571383 |

318 | 32488293351466654 | 2,13,3881,321965922259 |

319 | 34800954869440830 | 2,3,5,7,165718832711623 |

320 | 37274405776748077 | 7,109,48852432210679 |

321 | 39919565526999991 | 7,5702795075285713 |

322 | 42748078035954696 | 2,2,2,3,37,48139727517967 |

323 | 45772358543578028 | 2,2,47279,415343,582731 |

324 | 49005643635237875 | 5,5,5,17,19,1213762071461 |

325 | 52462044228828641 | 11,79,1013,38749,1537997 |

326 | 56156602112874289 | 13,701,6162251960153 |

327 | 60105349839666544 | 2,2,2,2,7,7,17,79,277,283,728207 |

328 | 64325374609114550 | 2,5,5,73,17623390303867 |

329 | 68834885946073850 | 2,5,5,523,661673,3978263 |

330 | 73653287861850339 | 3,24551095953950113 |

331 | 78801255302666615 | 5,463,701,1033,1069,43973 |

332 | 84300815636225119 | 29,31,151,37447,16583573 |

333 | 90175434980549623 | 29,67,149,2819,110492831 |

334 | 96450110192202760 | 2,2,2,5,7,7,43,61,86981,215687 |

335 | 103151466321735325 | 5,5,13,12161,26098933241 |

336 | 110307860425292772 | 2,2,3,11,41,341743,59641567 |

337 | 117949491546113972 | 2,2,20533,1436096668121 |

338 | 126108517833796355 | 5,11,13,176375549417897 |

339 | 134819180623301520 | 2,2,2,2,3,5,561746585930423 |

340 | 144117936527873832 | 2,2,2,3,3,3,109,6121217147803 |

341 | 154043597379576030 | 2,3,5,7,7,1079531,97071379 |

342 | 164637479165761044 | 2,2,3,3,19,8761,27473812231 |

343 | 175943559810422753 | 23,76253,178897,560771 |

344 | 188008647052292980 | 2,2,5,10937,617657,1391561 |

345 | 200882556287683159 | 79,113,1156801,19452617 |

346 | 214618299743286299 | 7,5923,28069,184416611 |

347 | 229272286871217150 | 2,3,3,3,5,5,11,2311,9257,721697 |

348 | 244904537455382406 | 2,3,7,17,23,14913197993873 |

349 | 261578907351144125 | 5,5,5,23,523,173965521557 |

350 | 279363328483702152 | 2,2,2,3,47,349,997,711770053 |

351 | 298330063062758076 | 2,2,3,7,106783,33259492333 |

352 | 318555973788329084 | 2,2,3391,382231,61442951 |

353 | 340122810048577428 | 2,2,3,239,118592332652921 |

354 | 363117512048110005 | 3,5,29,47,67,971,273002137 |

355 | 387632532919029223 | 7,24113,2296523706353 |

356 | 413766180933342362 | 2,101,3557,15199,37888267 |

357 | 441622981929358437 | 3,13,13,871051246408991 |

358 | 471314064268398780 | 2,2,3,3,3,5,11,11,13,401,1383705109 |

359 | 502957566506000020 | 2,2,5,101,349,104459,6829811 |

360 | 536679070310691121 | 11,14423,3382722484357 |

361 | 572612058898037559 | 3,3,131,485676046563221 |

362 | 610898403751884101 | 7,7,4793,2601150503293 |

363 | 651688879997206959 | 3,433,4793,104670363037 |

364 | 695143713458946040 | 2,2,2,5,43,4171819,96877003 |

365 | 741433159884081684 | 2,2,3,2137,89491,323077621 |

366 | 790738119649411319 | 790738119649411319 |

367 | 843250788562528427 | 599,1237,1138047093529 |

368 | 899175348396088349 | 73,101,179,382429,1781543 |

369 | 958728697912338045 | 3,5,7,11,751,1105283803889 |

370 | 1022141228367345362 | 2,49556849,10312814969 |

371 | 1089657644424399782 | 2,71,7673645383270421 |

372 | 1161537834849962850 | 2,3,3,5,5,7,547,757,890511641 |

373 | 1238057794119125085 | 3,5,683,722639,167227447 |

374 | 1319510599727473500 | 2,2,3,5,5,5,983,1831,5101,95813 |

375 | 1406207446561484054 | 2,3347,210069830678441 |

376 | 1498478743590581081 | 7,7,431,1949,66863,544477 |

377 | 1596675274490756791 | 2389,16197169,41263051 |

378 | 1701169427975813525 | 5,5,197,331031,1043452463 |

379 | 1812356499739472950 | 2,5,5,73,127,5081,769480909 |

380 | 1930656072350465812 | 2,2,11,43878547098874223 |

381 | 2056513475336633805 | 3,5,227,2801473,215589697 |

382 | 2190401332423765131 | 3,11,66375797952235307 |

383 | 2332821198543892336 | 2,2,2,2,7,7,53,227,177043,1396963 |

384 | 2484305294265418180 | 2,2,5,31,997,4019001026087 |

385 | 2645418340688763701 | 43,11738107,5241165101 |

386 | 2816759503217942792 | 2,2,2,67,2333,8963,12421,20233 |

387 | 2998964447736452194 | 2,131,1607,14149,14449,34841 |

388 | 3192707518433532826 | 2,1596353759216766413 |

389 | 3398704041358160275 | 5,5,13,43,20921,11624628749 |

390 | 3617712763867604423 | 7,7,25439,2902271029993 |

391 | 3850538434667429186 | 2,11,37717,4640466482039 |

392 | 4098034535626594791 | 3,3,137,167,619,32151809299 |

393 | 4361106170762284114 | 2,367,569,1283,21487,378779 |

394 | 4640713124699623515 | 3,5,17,359,78401,646589267 |

395 | 4937873096788191655 | 5,6553,46451,3244402177 |

396 | 5253665124416975163 | 3,3,49789639,11724137413 |

397 | 5589233202595404488 | 2,2,2,7,23,31,37,3783318894683 |

398 | 5945790114707874597 | 3,3,660643346078652733 |

399 | 6324621482504294325 | 3,3,5,5,191,147169784351467 |

400 | 6727090051741041926 | 2,23869,140916880718527 |

401 | 7154640222653942321 | 71,2381,151273,279774827 |

402 | 7608802843339879269 | 3,11,7151,32243012604043 |

403 | 8091200276484465581 | 7,3911,10093,29282410321 |

404 | 8603551759348655060 | 2,2,5,7,61453941138204679 |

405 | 9147679068859117602 | 2,3,34286363,44467043009 |

Quote:gordonm888Sorry, I spent some time looking through my many spreadsheets and I didn't find the work that I had mentioned. I did find some spreadsheets with the primes from 3-19,997 converted into prime number radices (bases) so I used them to produce the (modest) results shown in the table below...

Thank you. I can't think of a good comment offhand.

Partitions in a linear sequence

It is well known that if you have 5 objects that you can partition them into these seven arrangements:

1-1-1-1-1; 2-1-1-1; 2-2-1; 3-1-1; 3-2; 4-1; 5

Now let’s consider the 13 ranks in a standard deck of cards, and for the moment let’s define an Ace to be a high card only. (Equivalently, we could define an ace as a low card only with identical results)

Given: All the various combinations of 5 cards of different rank, and ignoring the suits of the cards

Defining: a 5 card straight as 5 consecutive ranks, a 4 card straight as 4 consecutive ranks, etc. ranging down to a 1-card straight which is a card that has a rank with both adjacent ranks empty, let’s look at some combinations of 5 different ranks

Ex: QJ962. Graphically, this hand looks like this: __QJ_9__6___2. It is easy to see that it has one 2- card straight and three 1-card straights. An obvious way to label this hand (from a connect-ness or straightness point of view) is 2-1-1-1

Ex: T7654. Graphically, this hand is ____T__7654__. It is one 4-card straight and one 1-card straight, which we label as 4-1.

Clearly, all the “straight patterns” are equivalent to the partitions of 5, because we are literally filling 5 of 13 ‘slots’ and noting how the 5 objects are partitioned.

Again, in the cards analogy I am defining aces as either high or low, but not both. However, we can define a mathematical relationship that is more general than cards:

Define a linear string(or array) of 13 slots such that each of the 11 interior slots is each connected to two adjacent slots and such that the two end slots are only connected to one adjacent slot each. Now consider every possible combination of ways to populate the 13 slots with 5 objects. There will be c(13,5) =1287 different combinations of ways to populate the 13 slots with 5 objects.

Surprisingly, given this definition, we can calculate the number of combinations that correspond to the various partitions of 5, equivalently we say that we can calculate the frequencies or probability densities of the partitions of 5 when in a linear string (without loops) of 13 spaces.

Case | 1-1-1-1-1 | 2-1-1-1 | 2-2-1 | 3-1-1 | 3-2 | 4-1 | 5 |
---|---|---|---|---|---|---|---|

5 into 13, line | 126 | 504 | 252 | 252 | 72 | 72 | 9 |

Of course, there is nothing fundamental about having a string length of 13. Here are some combination frequencies for partitions of 5 when the length of the string is 8-13 available slots.

Case | 1-1-1-1-1 | 2-1-1-1 | 2-2-1 | 3-1-1 | 3-2 | 4-1 | 5 |
---|---|---|---|---|---|---|---|

5 into 13, line | 126 | 504 | 252 | 252 | 72 | 72 | 9 |

5 into 12, line | 56 | 280 | 168 | 168 | 56 | 56 | 8 |

5 into 11, line | 21 | 140 | 105 | 105 | 42 | 42 | 7 |

5 into 10, line | 6 | 62 | 58 | 60 | 30 | 30 | 6 |

5 into 9, line | 1 | 20 | 30 | 30 | 20 | 20 | 5 |

5 into 8, line | 0 | 4 | 12 | 12 | 12 | 12 | 4 |

A couple of observations based on the above table:

1. The partitions 4-1 and 3-2 appear to be equally likely when 5 objects are placed in a linear string.

2. The “5 into 8” case shows an uncanny symmetry in the number of combinations for the partitions. I have yet to see anything that matches it in the various cases I have analyzed.

Partitions in a Closed Linear String

Consider the game Clock Solitaire where all 13 ranks are arrayed a circle. Further, define the existence of a “straight” in this circular configuration to include KA2 as a 3-card straight, QKA23 as a 5-card straight, etc. More generally consider a closed linear string such that the ends of the string are adjacent (as in a loop or circle) and such that every grid spot in the string has two connections; i.e. no ends.

Given this configuration, the partition frequencies are different as is shown in the table below.

Case | 1-1-1-1-1 | 2-1-1-1 | 2-2-1 | 3-1-1 | 3-2 | 4-1 | 5 |
---|---|---|---|---|---|---|---|

5 into 13, open line | 141 | 497 | 244 | 252 | 72 | 72 | 9 |

5 into 13, closed (loop) line | 92 | 455 | 272 | 274 | 91 | 90 | 13 |

The difference in these two cases has been illustrated by geometric differences in the configuration of the linear array, but the essential differences in the case arise not from geometry but from the definitions of adjacency or connect-ness.

Partitions of 3

The number 3 has the following three partitions: 1-1-1, 2-1, and 3. It’s a very simple set of partitions, but a good starting point for becoming familiar with this kind of analysis For game analysts, remembering that Ace is either high only or low only, this is “You are dealt 3 unpaired cards, how often do you have a 3-card straight, a 2-card straight, or no connected ranks at all (1-1-1).”

Case | 1-1-1 | 2-1 | 3 | Total |
---|---|---|---|---|

3 of 13, line | 165 | 110 | 11 | 286 |

3 of 12, line | 120 | 90 | 10 | 220 |

3 of 11, line | 84 | 72 | 9 | 165 |

3 of 10, line | 56 | 56 | 8 | 120 |

3 of 9, line | 35 | 42 | 7 | 84 |

3 of 8, line | 20 | 30 | 6 | 56 |

3 of 7, line | 10 | 20 | 5 | 35 |

3 of 6, line | 4 | 12 | 4 | 20 |

Now, let’s look at the those same results for partitions of three when the line is a closed loop (no end points)

Case | 1-1-1 | 2-1 | 3 | Total |
---|---|---|---|---|

3 of 13, closed loop | 157 | 116 | 13 | 286 |

3 of 12, closed loop | 112 | 97 | 12 | 220 |

3 of 11, closed loop | 77 | 77 | 11 | 165 |

3 of 10, closed loop | 50 | 60 | 10 | 120 |

3 of 9, closed loop | 30 | 45 | 9 | 84 |

3 of 8, closed loop | 16 | 32 | 8 | 56 |

3 of 7, closed loop | 7 | 21 | 7 | 35 |

3 of 6, closed loop | 2 | 12 | 6 | 20 |

There appears to be a lot of symmetry in this closed loop version of partitions of 3. At this point, I have not analyzed closed loop systems very much because I have become intrigued by some aspects of possible applications of the “open line” partition frequencies.

Partitions of 4.

The number 4 has the following five partitions: 1-1-1-1, 2-1-1, 2-2, 3-1, and 4. Just for chuckles, let’s see the frequencies of these partitions for open linear arrays of length 8-13.

Case | 1-1-1-1 | 2-1-1 | 2-2 | 3-1 | 4 | Total |
---|---|---|---|---|---|---|

4 of 13, line | 210 | 360 | 45 | 90 | 10 | 715 |

4 of 12, line | 126 | 252 | 36 | 72 | 9 | 495 |

4 of 11, line | 70 | 168 | 28 | 56 | 8 | 330 |

4 of 10, line | 35 | 105 | 21 | 42 | 7 | 210 |

4 of 9, line | 15 | 60 | 15 | 30 | 6 | 126 |

4 of 8, line | 5 | 30 | 10 | 20 | 5 | 70 |

I always like to look for patterns in numbers, look for primes and special numbers. But when I first calculated these partition frequencies/combinations they frustrated me. The only prime numbers are trivial. I learned to dig into them deeper, which I’ll discuss later in the next few posts.

I will explain an efficient way of calculating the frequency of any partition of the number m when chosen from an open linear array of n objects.

Let me define an example based on card games. Example: You are designing a bonus payout for a card game in which a player is dealt 7 cards and in which the Ace counts only as a high or low card and in which you are ignoring flushes. You want to evaluate a bonus payout for the player when his 7-card hand has no pairs and has two straights that are exactly 3-cards long. There are c(13,7) = 1716 possible combinations of 7 ranks, but how often will 7 cards of different ranks be partitioned into a 3-3-1 partition?

In order to implement the methodology I will describe, we must define for any given partition a TOTAL, a LENGTH and the NUMBER OF PERMUTATIONS

For 3-3-1, The total is 3+3+1=7. The length is 3, because this partition has 3 elements or clumps. And the number of permutations is 3, because it can be sequenced as followed: 3-3-1; 3-1-3; and 1-3-3.

Of specific importance to this calculation is the partition 3-3-1 has 3 permutations.

The next step is to examine the partition of the spaces between the 3 substrings that make up 3-3-1. I call this “partitioning the void.” When selecting 7 objects from a total of 13, you can define the 7 objects that were selected equally well by defining the 6 objects that you have not selected. When picking 7 card ranks from 13, we can equally well think of this as defining 6 card ranks that are NOT SELECTED. This is the reason that c(13,7) =c(13,6) and more generally, that

Now, given 7 selected from 13, and given the 7 objects will be partitioned as 3-3-1, we know that there must be 6 objects that were not selected and the 6 unselected objects must have a partition that has a length in the range of 2-4. Clearly, the unselected objects must have at least two ‘clumps ‘ so as to divide the 7 objects into three clumps. Similarly, if the 6 unselected objects were partitioned into 5 or more clumps (2-1-1-1, or 1-1-1-1-1-1) they would have to divide the 7 selected objects into more than 3 clumps.

So given an open linear string of n objects and selecting m objects with a partition of length l, the unselected objects must total (n-m) and be arrayed as a partition of (n-m) with length in the rangel -1 to l+1.

For our example, which involves 7 ranks selected from 13 and arrayed as a partition of 3-3-1, let’s look at the partitions of 6. We’ll order them by their length:

l=5: 2-1-1-1-1

l=4: 3-1-1-1; 2-2-1-1

l=3: 4-1-1; 3-2-1; 2-2-2

l=2: 5-1; 4-2; 3-3

l=1: 6

For our example of 7 ranks selected from 13 and arrayed as a partition of 3-3-1, the 6 unselected ranks must be partitioned as either 3-1-1-1; 2-2-1-1, 4-1-1; 3-2-1; 2-2-2, 5-1; 4-2; 3-3 because these are all the possible partitions of 6 that have a length of 2-4.

Now, I define a parameter that will be useful in some applications.

p(n,l) = the sum of the number of permutations for all partitions of n with length l.

Example p(6,3) is the sum of number of permutations for all partitions of 6 with length =3. Referring to the list above we have:

4-1-1 which has 3 permutations

3-2-1 which has 6 permutations

2-2-2 which has 1 permutation

So, p(6,3) = 3+6+1 =10.

Similarly, p(6,6)=1; p(6,5)=5; p(6,4)=10: p(6,2)=5 and p(6,1)=1.

The number of combinations of 7 ranks picked out of 13 that have a partition of 3-3-1 I is the product:

So, when selecting 7 ranks out of 13 (and restricting the ace to be either high or low) the 7 ranks will be partitioned as 3-3-1 in 105 of 1716 possible combinations.

It is straightforward to calculate values of p(n,l) for small values of n; I have found that it is useful to have precalculated tables of this parameter when working certain kinds of problems.

The binary form of integers is an obvious opportunity to investigate partitions in open linear strings. I have just started to consider this, but here are some preliminary examples and observations.

The decimal integer 137 may be written in binary notation as 10001001. Now notice that 10001001 is a string of 8 digits: 3 ones and 5 zeros. The ones are partitioned as 1-1-1 and the 5 zeros are partitioned as 3-2.

The partitions 1-1-1|3-2 do not uniquely connote 197, there are 2 binary numbers with those partitions. We know it is 2 because 1-1-1 has 1 permutation and 3-2 has 2 permutations and 1x2=2.

The set of binary integers with partitions 1-1-1|3-1 is (145, 197).

As another example, the set of binary integers with partitions 2-1-1|2-1 is (75, 77, 83, 89, 101, 105). Here are some other sets, as denoted by combinations of simple partitions.

2-1|1 = (11, 13)

3-1|1 = (23, 29)

4-1|1 = (47, 61)

5-1|1 = (95, 125)

6-1|1 = (191, 253)

2-1|2 = (19, 25)

3-1|3 = (71, 113)

4-1|3 = (143, 241)

2-1-1|1-1 = (43, 45, 53)

Here are some rules for interpreting the partition nomenclature:

1. The first partition is the partition of the 1’s, the second partition refers to the partition of the 0’s.

2. The number of binary integers that have that specific set of partitions for the 1s and 0s will be equal to the product of the permutations of the two partitions.

3. In order to be a valid binary integer, the length of the zeros partition must be equal to or one less than the length of the ones partition.

4. If the length of the partition of zeros is one less than the length of the partition of the 1s, than all the integers in the set will be ODD. If the length of the two partitions are equal, then all the integers in the set will be EVEN.

So given this symbol: 4-3-1-1|2-2-1 it can be immediately deduced that 36 integers will have that configuration of partitions, and they will all be odd numbers in the range 2

^{14}to 2

^{15-1}.

Now let’s look at partitions when writing binary numbers with some of the leading zeroes. To do this it is necessary to define a number “space” or region. Let’s consider all the integers from 0 to (n

^{13}-1) and write them all as having 13 digits.

8,191 = 11111111111111 which is 13|0

7,621 = 1110111000101 which is 3-3-1-1|3-1-1

Notice that for 7,621 the ones partition has a length l=4, while the zeroes partition has a length that is 3, or l-1. Here again, we see that if the ones partition has a length n then the length of the ones partition is constrained to be n-1, n, or n+1.

Let’s make a definition that in the space 0 to (2

^{13}-1) that a number is:

SMALL if it is 0 to (2

^{12}-1)

LARGE if it is 2

^{12}to (2

^{12}-1)

Now if the ones partition of an integer in the space 0 to (2

^{13}-1) is of length l, then it is easily proven that:

- The integer will be SMALL and ODD if the zeros partition is of length ( l - 1 )

- The integer will be LARGE and EVEN if the zeros partition is of length ( l + 1 )

- The integer will be either SMALL and EVEN or LARGE and ODD if the zeros partition is of length l

Combinations and Factor of Two

Let me write down one particular mathematical identity that came to me as I was working with binary numbers.

_{m=1 to n}Σ c(n,m) = 2

^{n}– 1

where, again, c(n,m) is the classic formula for combinations when selecting m from n.

Example: Take n=13

c(13,13) =1

c(13,1) = c(13,12) = 13

c(13,2) = c(13,11) =78

c(13,3) = c(13,10) =286

c(13,4) = c(13,9) =715

c(13,5) = c(13,8) =1287

c(13,6) = c(13,7) =1716

and:

1+ 2 x (13 + 78 + 286 + 715 + 1287 + 1716) = 8,191 = 2

^{13}- 1

This simple formula potentially links combination math to other areas of number theory such as Mersenne Primes.

The identity may also be written as:

_{m=0 to n}Σc(n,m) = 2

^{n}

Where now the summation starts at m=0.

Interesting, huh?