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Wizard
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Wizard
Joined: Oct 14, 2009
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Thanks for this post from:
Zuga
August 14th, 2019 at 12:39:04 AM permalink
I've been thinking about partions off and on lately. If the term sounds familiar, it was at the center of the movie The Man Who Knew Infinity.

If you didn't see the movie, it is the number of ways you can break down an integer into other integers of lesser or equal size. For example, let's say a blackjack dealer has chips in denominations from $1 to $5. You ask to break down a $5 chip and she asks "how?" How many ways can it be done?

The answer is 7:

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

Granted, getting another $5 chip back isn't breaking it down, but to be consistent, it counts as a partition.

I wrote both a spreadsheet and a program to get at answers for large numbers. Here they are for 1 to 405. At 406, I exceed the size of a 64-bit unsigned integer.



TotalPartitions
11
22
33
45
57
611
715
822
930
1042
1156
1277
13101
14135
15176
16231
17297
18385
19490
20627
21792
221002
231255
241575
251958
262436
273010
283718
294565
305604
316842
328349
3310143
3412310
3514883
3617977
3721637
3826015
3931185
4037338
4144583
4253174
4363261
4475175
4589134
46105558
47124754
48147273
49173525
50204226
51239943
52281589
53329931
54386155
55451276
56526823
57614154
58715220
59831820
60966467
611121505
621300156
631505499
641741630
652012558
662323520
672679689
683087735
693554345
704087968
714697205
725392783
736185689
747089500
758118264
769289091
7710619863
7812132164
7913848650
8015796476
8118004327
8220506255
8323338469
8426543660
8530167357
8634262962
8738887673
8844108109
8949995925
9056634173
9164112359
9272533807
9382010177
9492669720
95104651419
96118114304
97133230930
98150198136
99169229875
100190569292
101214481126
102241265379
103271248950
104304801365
105342325709
106384276336
107431149389
108483502844
109541946240
110607163746
111679903203
112761002156
113851376628
114952050665
1151064144451
1161188908248
1171327710076
1181482074143
1191653668665
1201844349560
1212056148051
1222291320912
1232552338241
1242841940500
1253163127352
1263519222692
1273913864295
1284351078600
1294835271870
1305371315400
1315964539504
1326620830889
1337346629512
1348149040695
1359035836076
13610015581680
13711097645016
13812292341831
13913610949895
14015065878135
14116670689208
14218440293320
14320390982757
14422540654445
14524908858009
14627517052599
14730388671978
14833549419497
14937027355200
15040853235313
15145060624582
15249686288421
15354770336324
15460356673280
15566493182097
15673232243759
15780630964769
15888751778802
15997662728555
160107438159466
161118159068427
162129913904637
163142798995930
164156919475295
165172389800255
166189334822579
167207890420102
168228204732751
169250438925115
170274768617130
171301384802048
172330495499613
173362326859895
174397125074750
175435157697830
176476715857290
177522115831195
178571701605655
179625846753120
180684957390936
181749474411781
182819876908323
183896684817527
184980462880430
1851071823774337
1861171432692373
1871280011042268
1881398341745571
1891527273599625
1901667727404093
1911820701100652
1921987276856363
1932168627105469
1942366022741845
1952580840212973
1962814570987591
1973068829878530
1983345365983698
1993646072432125
2003972999029388
2014328363658647
2024714566886083
2035134205287973
2045590088317495
2056085253859260
2066622987708040
2077206841706490
2087840656226137
2098528581302375
2109275102575355
21110085065885767
21210963707205259
21311916681236278
21412950095925895
21514070545699287
21615285151248481
21716601598107914
21818028182516671
21919573856161145
22021248279009367
22123061871173849
22225025873760111
22327152408925615
22429454549941750
22531946390696157
22634643126322519
22737561133582570
22840718063627362
22944132934884255
23047826239745920
23151820051838712
23256138148670947
23360806135438329
23465851585970275
23571304185514919
23677195892663512
23783561103925871
23890436839668817
23997862933703585
240105882246722733
241114540884553038
242123888443077259
243133978259344888
244144867692496445
245156618412527946
246169296722391554
247182973889854026
248197726516681672
249213636919820625
250230793554364681
251249291451168559
252269232701252579
253290726957916112
254313891991306665
255338854264248680
256365749566870782
257394723676655357
258425933084409356
259459545750448675
260495741934760846
261534715062908609
262576672674947168
263621837416509615
264670448123060170
265722760953690372
266779050629562167
267839611730366814
268904760108316360
269974834369944625
2701050197489931117
2711131238503938606
2721218374349844333
2731312051800816215
2741412749565173450
2751520980492851175
2761637293969337171
2771762278433057269
2781896564103591584
2792040825852575075
2802195786311682516
2812362219145337711
2822540952590045698
2832732873183547535
2842938929793929555
2853160137867148997
2863397584011986773
2873652430836071053
2883925922161489422
2894219388528587095
2904534253126900886
2914872038056472084
2925234371069753672
2935622992691950605
2946039763882095515
2956486674127079088
2966965850144195831
2977479565078510584
2988030248384943040
2998620496275465025
3009253082936723602
3019930972392403501
30210657331232548839
30311435542077822104
30412269218019229465
30513162217895057704
30614118662665280005
30715142952738857194
30816239786535829663
30917414180133147295
31018671488299600364
31120017426762576945
31221458096037352891
31323000006655487337
31424650106150830490
31526415807633566326
31628305020340996003
31730326181989842964
31832488293351466654
31934800954869440830
32037274405776748077
32139919565526999991
32242748078035954696
32345772358543578028
32449005643635237875
32552462044228828641
32656156602112874289
32760105349839666544
32864325374609114550
32968834885946073850
33073653287861850339
33178801255302666615
33284300815636225119
33390175434980549623
33496450110192202760
335103151466321735325
336110307860425292772
337117949491546113972
338126108517833796355
339134819180623301520
340144117936527873832
341154043597379576030
342164637479165761044
343175943559810422753
344188008647052292980
345200882556287683159
346214618299743286299
347229272286871217150
348244904537455382406
349261578907351144125
350279363328483702152
351298330063062758076
352318555973788329084
353340122810048577428
354363117512048110005
355387632532919029223
356413766180933342362
357441622981929358437
358471314064268398780
359502957566506000020
360536679070310691121
361572612058898037559
362610898403751884101
363651688879997206959
364695143713458946040
365741433159884081684
366790738119649411319
367843250788562528427
368899175348396088349
369958728697912338045
3701022141228367345362
3711089657644424399782
3721161537834849962850
3731238057794119125085
3741319510599727473500
3751406207446561484054
3761498478743590581081
3771596675274490756791
3781701169427975813525
3791812356499739472950
3801930656072350465812
3812056513475336633805
3822190401332423765131
3832332821198543892336
3842484305294265418180
3852645418340688763701
3862816759503217942792
3872998964447736452194
3883192707518433532826
3893398704041358160275
3903617712763867604423
3913850538434667429186
3924098034535626594791
3934361106170762284114
3944640713124699623515
3954937873096788191655
3965253665124416975163
3975589233202595404488
3985945790114707874597
3996324621482504294325
4006727090051741041926
4017154640222653942321
4027608802843339879269
4038091200276484465581
4048603551759348655060
4059147679068859117602



I'll put my code in spoiler tags, if anyone wants to tackle this as a programming exercise.




void partitions(int max_series)
{
int i,j,k;
unsigned __int64 tot;
for (i=0; i<=1000; i++)
for (j=0; j<=1000; j++)
partition_array
=0;
partition_array[1][0]=1;
for (i=2; i<=max_series; i++)
{
for (j=1; j<=i; j++) // first column
{
if (j*2<i) // less than half of total
{
tot=0;
for (k=1; k<=j; k++)
tot+=partition_array[i-j][k];
partition_array
=tot;
}
else if (j==i)
partition_array
=1;
else
partition_array
=partition_array[i-j][0];
partition_array[0]+=partition_array
;
}
printf("%i\t%I64i\n",i,partition_array[0]);
}
}



All that said, I'm not sure what my question is. Has anyone else messed with this? Please don't just plop in a Numberfile video and call it a day, let's try to have a discussion.

One of the mysteries of mathematics is a simple formula to the number of partitions of x, that doesn't just sum partitions of smaller numbers (as I did). As the movie I linked to goes into, Srinivasa Ramanujan found some relationships between a number and its number of partitions.

The question for the poll is how do you feel about partitions?
It's not whether you win or lose; it's whether or not you had a good bet.
AcesAndEights
AcesAndEights
Joined: Jan 5, 2012
  • Threads: 66
  • Posts: 4261
August 14th, 2019 at 4:49:41 AM permalink
A similar concept is sometimes used as an interview question for software development roles. "Given standard USA coins, write a function that will take a dollar amount and return the number of different ways you can make that dollar amount in coins." I think I tried to solve it with recursion, but iteration would work too. Your function is also printing out the combinations which wasn't a requirement the one time I was asked this.

Because I hate interviewing, I don't feel good about partitions :P
"So drink gamble eat f***, because one day you will be dust." -ontariodealer
rsactuary
rsactuary 
Joined: Sep 6, 2014
  • Threads: 15
  • Posts: 1408
Thanks for this post from:
tringlomane
August 14th, 2019 at 7:26:08 AM permalink
I've been working on math behind the Grand Jackpot for the Lightning Link / Dragon Link games and I had to research and use partitions in that. Crazy that you mention it right now, because I'm about ready to share my work if anyone is interested.
Ayecarumba
Ayecarumba
Joined: Nov 17, 2009
  • Threads: 229
  • Posts: 6455
August 14th, 2019 at 10:53:38 AM permalink
Not sure how helpful this is, but here is a link to an article on Discovery.com that indicates something is there waiting to be described.
Simplicity is the ultimate sophistication - Leonardo da Vinci
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1210
  • Posts: 20160
Thanks for this post from:
tringlomane
August 14th, 2019 at 11:06:56 AM permalink
Quote: rsactuary

I'm about ready to share my work if anyone is interested.



I'm interested.
It's not whether you win or lose; it's whether or not you had a good bet.
rsactuary
rsactuary 
Joined: Sep 6, 2014
  • Threads: 15
  • Posts: 1408
August 14th, 2019 at 11:32:20 AM permalink
PM me an email address? We might need to get on the phone for me to walk you through it. It's a LOT of math.
tringlomane
tringlomane
Joined: Aug 25, 2012
  • Threads: 8
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August 14th, 2019 at 9:21:28 PM permalink
Quote: Wizard

I'm interested.



I would also be interested.
gordonm888
gordonm888
Joined: Feb 18, 2015
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Thanks for this post from:
tringlomanecharliepatrick
August 14th, 2019 at 9:37:54 PM permalink
I have recently been doing some analytical/theoretical work on partitions, although going in a completely different direction than you have. I'm writing it up in a way suitable for WOV and I'll try to post some of it in a day or two and hope for some feedback.
Sometimes, people are just a bottomless mystery. And, after all, this is just a sh*tty little forum in the sun-less backwaters of the online world.
rsactuary
rsactuary 
Joined: Sep 6, 2014
  • Threads: 15
  • Posts: 1408
Thanks for this post from:
tringlomane
August 14th, 2019 at 10:09:25 PM permalink
Quote: tringlomane

I would also be interested.



Let me run it by Wiz and see what he thinks. I stress that it is strictly a theoretical exercise as there are too many unknowns without the PAR sheets to be able to apply it.
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1210
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August 15th, 2019 at 9:33:30 AM permalink
I noticed the total number of partitions is often prime or primey* . Just look at the first five total partitions: 2,3,5,7,11.

That caused me to factor the number of partitions to see how primey they were.


n Partitions of n Factorization of Partitions of n
222
333
455
577
61111
7153,5
8222,11
9302,3,5
10422,3,7
11562,2,2,7
12777,11
13101101
141353,3,3,5
151762,2,2,2,11
162313,7,11
172973,3,3,11
183855,7,11
194902,5,7,7
206273,11,19
217922,2,2,3,3,11
2210022,3,167
2312555,251
2415753,3,5,5,7
2519582,11,89
2624362,2,3,7,29
2730102,5,7,43
2837182,11,13,13
2945655,11,83
3056042,2,3,467
3168422,11,311
3283493,11,11,23
33101433,3,7,7,23
34123102,5,1231
35148833,11,11,41
361797717977
37216377,11,281
38260155,11,11,43
39311853,3,3,3,5,7,11
40373382,3,7,7,127
41445833,7,11,193
42531742,11,2417
43632613,3,3,3,11,71
44751755,5,31,97
45891342,41,1087
461055582,3,73,241
471247542,7,7,19,67
481472733,7,7013
491735255,5,11,631
502042262,11,9283
512399433,11,11,661
522815893,7,11,23,53
533299313,3,7,5237
543861555,7,11,17,59
554512762,2,7,71,227
5652682311,47,1019
576141542,3,102359
587152202,2,5,11,3251
598318202,2,5,11,19,199
6096646717,139,409
6111215053,5,7,11,971
6213001562,2,11,13,2273
6315054993,113,4441
6417416302,5,11,71,223
6520125582,1006279
6623235202,2,2,2,2,2,5,53,137
6726796891181,2269
6830877353,5,7,7,4201
6935543455,641,1109
7040879682,2,2,2,2,3,97,439
7146972053,5,313147
72539278311,139,3527
73618568923,131,2053
7470895002,2,5,5,5,11,1289
7581182642,2,2,3,7,11,23,191
7692890917,1327013
771061986310619863
78121321642,2,11,103,2677
79138486502,5,5,173,1601
80157964762,2,3,3,227,1933
811800432711,1636757
82205062555,7,7,7,11,1087
83233384697,11,303097
84265436602,2,5,11,13,9281
853016735711,11,249317
86342629622,23,37,41,491
873888767311,3535243
88441081093,3,83,137,431
89499959255,5,7,7,40813
90566341732473,22901
916411235929,373,5927
92725338073371,21517
938201017759,1390003
94926697202,2,2,5,11,13,17,953
95104651419283,369793
961181143042,2,2,2,2,2,2,2,2,2,2,7,7,11,107
971332309302,3,5,7,29,131,167
981501981362,2,2,11,1706797
991692298755,5,5,1353839
1001905692922,2,43,59,89,211
1012144811262,31,3459373
1022412653793,2423,33191
1032712489502,5,5,7,774997
1043048013653,5,11,1847281
10534232570911,43,43,16831
1063842763362,2,2,2,3,8005757
1074311493893,11,173,75521
1084835028442,2,11,10988701
1095419462402,2,2,2,2,2,2,5,11,23,3347
1106071637462,7,4049,10711
1116799032033,7,67,483229
1127610021562,2,190250539
1138513766282,2,212844157
1149520506655,193,986581
11510641444513,61,67,229,379
11611889082482,2,2,11,11,157,7823
11713277100762,2,7,7,11,615821
118148207414311,197,827,827
11916536686655,11,30066703
12018443495602,2,2,5,47,981037
1212056148051461,4460191
12222913209122,2,2,2,9013,15889
123255233824179,32308079
12428419405002,2,3,5,5,5,7,31,8731
12531631273522,2,2,3,7,11,59,67,433
12635192226922,2,89,379,26083
12739138642955,11,67,1062107
12843510786002,2,2,5,5,11,17,317,367
12948352718702,3,3,5,11,13,157,2393
13053713154002,2,2,5,5,11,157,15551
13159645395042,2,2,2,7,7,7,11,29,3407
13266208308896620830889
13373466295122,2,2,3,3,1319,77359
13481490406955,17,89,1077203
13590358360762,2,59,569,67289
136100155816802,2,2,2,5,13,31,41,7577
137110976450162,2,2,17,1367,59693
138122923418313,3,7,7,11,733,3457
139136109498955,79,34458101
140150658781355,7,11,39132151
141166706892082,2,2,29,31,991,2339
142184402933202,2,2,3,3,5,127,403331
143203909827577589,2686913
144225406544455,5807,776327
145249088580097,7,53,73,83,1583
1462751705259953197,517267
147303886719782,3,7,24151,29959
1483354941949711,73,41780099
149370273552002,2,2,2,2,2,5,5,11,11,11,17387
1504085323531311,17,197,1108967
151450606245822,3,7510104097
152496862884217,11,751,859223
153547703363242,2,11,34513,36067
154603566732802,2,2,2,2,2,2,2,5,37,1274423
1556649318209719,8087,432749
15673232243759463,1777,89009
1578063096476980630964769
158887517788022,79,691,853,953
159976627285555,7,29,67,1436111
1601074381594662,3,3,11,443,1224869
161118159068427797,148254791
1621299139046373,3,11,127,10332769
1631427989959302,3,5,4759966531
1641569194752953,3,5,11,14867,21323
1651723898002555,313,1543,71389
1661893348225797,7,37,53,1277,1543
1672078904201022,19,73,503,148991
168228204732751228204732751
1692504389251153,3,3,5,1855103149
1702747686171302,5,7,47,83516297
1713013848020482,2,2,2,2,2,2,2,11,11,1609,6047
172330495499613103,2351,1364821
1733623268598953,5,7,11,13,37,652189
1743971250747502,5,5,5,103,1627,9479
1754351576978302,5,43515769783
1764767158572902,5,443,107610803
1775221158311955,7,97,153789641
1785717016056553,5,17,2241967081
1796258467531202,2,2,2,2,5,277,3467,4073
1806849573909362,2,2,3,7,7,13,59,643,1181
1817494744117813,249824803927
18281987690832311,23,27967,115873
18389668481752761,293,50169799
1849804628804302,5,11,107,83301859
18510718237743373469,6653,46441
18611714326923731171432692373
18712800110422682,2,7,7,18713,348991
18813983417455711398341745571
18915272735996253,3,3,3,5,5,5,61,127,19471
1901667727404093317,7283,722363
19118207011006522,2,3,7,11171,1940293
192198727685636323,503,171776027
19321686271054693,11,71,269,3440807
19423660227418455,7,7,521,18535961
19525808402129733,11,78207279181
19628145709875913,11,89,958314943
19730688298785302,5,13,43,257,2136131
19833453659836982,3,3,185853665761
19936460724321253,3,5,5,5,3240953273
20039729990293882,2,3,331083252449
20143283636586477,19,23,47,4441,6779
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2035134205287973151,34001359523
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20560852538592602,2,5,304262692963
20666229877080402,2,2,5,11,15052244791
20772068417064902,3,3,5,829,5399,17891
20878406562261373,7,373364582197
20985285813023753,5,5,5,19,197,2083,2917
21092751025753555,487,1091,3491363
211100850658857673,17,61,227,14280811
2121096370720525910963707205259
213119166812362782,7,13,31,79,131,409,499
214129500959258953,5,2351,7573,48491
215140705456992877,7,11,11,23497,100999
2161528515124848115285151248481
217166015981079142,2140069,3878753
2181802818251667153,340154387107
219195738561611455,11,13,313,87463331
220212482790093673,7082759669789
221230618711738493,223,138283,249287
222250258737601113,7,17,1259,55679497
223271524089256153,5,23,353,4801,46439
224294545499417502,5,5,5,16301,7227667
225319463906961573,7,37,97,2887,146819
226346431263225193,11,13,13,79,78630193
227375611335825702,5,13,173,449,3719657
228407180636273622,11,23,80470481477
229441329348842555,7,7,191,943112189
230478262397459202,2,2,2,2,2,2,5,109843,680321
231518200518387122,2,2,3,59,59,620272573
232561381486709473,7,32009,83515423
23360806135438329307,347,9749,58549
234658515859702755,5,7,376294776973
235713041855149197,10186312216417
236771958926635122,2,2,7,7,263,748776797
2378356110392587111,11,13,17,677,4615703
238904368396688173,7591,3971230829
239978629337035853,5,1114697,5852887
2401058822467227333,13,13,83,2516153293
2411145408845530382,57270442276519
2421238884430772597,31,570914484227
2431339782593448882,2,2,7,7,97,5783,609289
2441448676924964455,14813,1955953453
2451566184125279462,4007,19543101139
2461692967223915542,431,18433,10654799
2471829738898540262,77899,1174430287
2481977265166816722,2,2,11,79,28441673861
2492136369198206253,3,3,5,5,5,5,7,277,6529121
2502307935543646813,7,11,10037,99542323
251249291451168559887,281050114057
2522692327012525794177,64455997427
2532907269579161122,2,2,2,53,73,4696416353
2543138919913066655,17,643,9649,595207
2553388542642486802,2,2,5,631,859,1429,10937
2563657495668707822,91381,2001234211
2573947236766553573,7,18796365555017
2584259330844093562,2,545863,195073253
2594595457504486753,5,5,11,234197,2378447
2604957419347608462,31,53,150864861461
26153471506290860911,41,349,20389,166619
2625766726749471682,2,2,2,2,19,439,2160534839
2636218374165096155,419,29867,9938051
2646704481230601702,5,7,7,13,41,1907,1346143
2657227609536903722,2,419,8167,52802941
2667790506295621673,3,3,28853727020821
2678396117303668142,3,53,6857,385050089
2689047601083163602,2,2,3,3,5,2513222523101
2699748343699446255,5,5,173,45079046009
27010501974899311173,11,83,47309,8104667
27111312385039386062,7,3041,26571111569
272121837434984433311,317333,349038091
27313120518008162155,499,2003,9829,26711
27414127495651734502,5,5,7,7,173,4327,770311
27515209804928511755,5,60839219714047
2761637293969337171167,1777,1931,2857199
27717622784330572693,3,29,6752024647729
27818965641035915842,2,2,2,2,3,7,7,7,23,2504230711
27920408258525750755,5,373,218855319311
28021957863116825162,2,7,389,3853,52321891
28123622191453377113,11,37,983809,1966499
28225409525900456982,29,59,619,1199570861
28327328731835475355,7,7,53,210463857031
28429389297939295553,5,790189,247951633
28531601378671489977,7,7,52517,175433287
28633975840119867733,3,3,7,43,547,764280217
287365243083607105373,50033299124261
28839259221614894222,3,103,2162249,2937971
28942193885285870953,5,263,1069553492671
29045342531269008862,3,227,9349,356093047
29148720380564720842,2,1218009514118021
29252343710697536722,2,2,7,7,11,97,223,56118901
29356229926919506055,7087043,158683747
29460397638820955155,11,13,73,1187,7027,13873
29564866741270790882,2,2,2,461,1033,851335711
29669658501441958313,7,83,5783,691072399
29774795650785105842,2,2,3,3,3413,6971,4366289
29880302483849430402,2,2,2,2,2,3,5,4801,1742312449
29986204962754650255,5,7,17,2897645806879
30092530829367236022,137,1021,33075784213
30199309723924035013,6491,509986771037
3021065733123254883910657331232548839
303114355420778221042,2,2,11,53,2451874373461
304122692180192294653,5,7,257,617,1249,589993
305131622178950577042,2,2,11,117101,1277279083
306141186626652800055,7,23,67,79,2357,1405841
307151429527388571942,3,3,41,607,33803799259
3081623978653582966311149,68473,21272819
309174141801331472955,12650773,275306183
310186714882996003642,2,181,503,51271070537
311200174267625769455,13,31,199,33479,1491103
312214580960373528917,15493,217027,911683
313230000066554873377,7,7,41,1635497877799
314246501061508304902,3,3,5,11,59,422018595289
315264158076335663262,691,1609,11879544977
316283050203409960033,3,11,113,233,10859101993
317303261819898429642,2,557,44111,308571383
318324882933514666542,13,3881,321965922259
319348009548694408302,3,5,7,165718832711623
320372744057767480777,109,48852432210679
321399195655269999917,5702795075285713
322427480780359546962,2,2,3,37,48139727517967
323457723585435780282,2,47279,415343,582731
324490056436352378755,5,5,17,19,1213762071461
3255246204422882864111,79,1013,38749,1537997
3265615660211287428913,701,6162251960153
327601053498396665442,2,2,2,7,7,17,79,277,283,728207
328643253746091145502,5,5,73,17623390303867
329688348859460738502,5,5,523,661673,3978263
330736532878618503393,24551095953950113
331788012553026666155,463,701,1033,1069,43973
3328430081563622511929,31,151,37447,16583573
3339017543498054962329,67,149,2819,110492831
334964501101922027602,2,2,5,7,7,43,61,86981,215687
3351031514663217353255,5,13,12161,26098933241
3361103078604252927722,2,3,11,41,341743,59641567
3371179494915461139722,2,20533,1436096668121
3381261085178337963555,11,13,176375549417897
3391348191806233015202,2,2,2,3,5,561746585930423
3401441179365278738322,2,2,3,3,3,109,6121217147803
3411540435973795760302,3,5,7,7,1079531,97071379
3421646374791657610442,2,3,3,19,8761,27473812231
34317594355981042275323,76253,178897,560771
3441880086470522929802,2,5,10937,617657,1391561
34520088255628768315979,113,1156801,19452617
3462146182997432862997,5923,28069,184416611
3472292722868712171502,3,3,3,5,5,11,2311,9257,721697
3482449045374553824062,3,7,17,23,14913197993873
3492615789073511441255,5,5,23,523,173965521557
3502793633284837021522,2,2,3,47,349,997,711770053
3512983300630627580762,2,3,7,106783,33259492333
3523185559737883290842,2,3391,382231,61442951
3533401228100485774282,2,3,239,118592332652921
3543631175120481100053,5,29,47,67,971,273002137
3553876325329190292237,24113,2296523706353
3564137661809333423622,101,3557,15199,37888267
3574416229819293584373,13,13,871051246408991
3584713140642683987802,2,3,3,3,5,11,11,13,401,1383705109
3595029575665060000202,2,5,101,349,104459,6829811
36053667907031069112111,14423,3382722484357
3615726120588980375593,3,131,485676046563221
3626108984037518841017,7,4793,2601150503293
3636516888799972069593,433,4793,104670363037
3646951437134589460402,2,2,5,43,4171819,96877003
3657414331598840816842,2,3,2137,89491,323077621
366790738119649411319790738119649411319
367843250788562528427599,1237,1138047093529
36889917534839608834973,101,179,382429,1781543
3699587286979123380453,5,7,11,751,1105283803889
37010221412283673453622,49556849,10312814969
37110896576444243997822,71,7673645383270421
37211615378348499628502,3,3,5,5,7,547,757,890511641
37312380577941191250853,5,683,722639,167227447
37413195105997274735002,2,3,5,5,5,983,1831,5101,95813
37514062074465614840542,3347,210069830678441
37614984787435905810817,7,431,1949,66863,544477
37715966752744907567912389,16197169,41263051
37817011694279758135255,5,197,331031,1043452463
37918123564997394729502,5,5,73,127,5081,769480909
38019306560723504658122,2,11,43878547098874223
38120565134753366338053,5,227,2801473,215589697
38221904013324237651313,11,66375797952235307
38323328211985438923362,2,2,2,7,7,53,227,177043,1396963
38424843052942654181802,2,5,31,997,4019001026087
385264541834068876370143,11738107,5241165101
38628167595032179427922,2,2,67,2333,8963,12421,20233
38729989644477364521942,131,1607,14149,14449,34841
38831927075184335328262,1596353759216766413
38933987040413581602755,5,13,43,20921,11624628749
39036177127638676044237,7,25439,2902271029993
39138505384346674291862,11,37717,4640466482039
39240980345356265947913,3,137,167,619,32151809299
39343611061707622841142,367,569,1283,21487,378779
39446407131246996235153,5,17,359,78401,646589267
39549378730967881916555,6553,46451,3244402177
39652536651244169751633,3,49789639,11724137413
39755892332025954044882,2,2,7,23,31,37,3783318894683
39859457901147078745973,3,660643346078652733
39963246214825042943253,3,5,5,191,147169784351467
40067270900517410419262,23869,140916880718527
401715464022265394232171,2381,151273,279774827
40276088028433398792693,11,7151,32243012604043
40380912002764844655817,3911,10093,29282410321
40486035517593486550602,2,5,7,61453941138204679
40591476790688591176022,3,34286363,44467043009


* To coin my own term, meaning to have rather few factors, compared to the size of the number. Is there a more technical term for this, and please don't say "relatively prime."
Last edited by: Wizard on Aug 15, 2019
It's not whether you win or lose; it's whether or not you had a good bet.

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