PPDIs

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Pluperfect Digital Invariants (PPDIs)


Note: In the table below, “O” stands for order (length). Every PPDI is a PDI. HPDIs are shown in red.
 

Base

O

Number (Representation)

Base-10 Value
2 1 0 0
2 1 1 1
3 1 0 0
3 1 1 1
3 1 2 2
3 2 12 5
3 2 22 8
3 3 122 17
4 1 0 0
4 1 1 1
4 1 2 2
4 1 3 3
4 3 130 28
4 3 131 29
4 3 203 35
4 3 223 43
4 3 313 55
4 3 332 62
4 4 1103 83
4 4 3303 243
5 1 0 0
5 1 1 1
5 1 2 2
5 1 3 3
5 1 4 4
5 2 23 13
5 2 33 18
5 3 103 28
5 3 433 118
5 4 2124 289
5 4 2403 353
5 4 3134 419
5 6 124030 4890
5 6 124031 4891
5 6 242423 9113
5 9 434434444 1874374
5 13 1143204434402 338749352
5 14 14421440424444 2415951874
6 1 0 0
6 1 1 1
6 1 2 2
6 1 3 3
6 1 4 4
6 1 5 5
6 3 243 99
6 3 514 190
6 5 14340 2292
6 5 14341 2293
6 5 14432 2324
6 5 23520 3432
6 5 23521 3433
6 5 44405 6197
6 6 435152 36140
6 7 5435254 269458
6 8 12222215 391907
6 9 555435035 10067135
6 13 1053025020422 2510142206
6 13 1053122514003 2511720147
6 13 1435403205450 3866632806
6 13 1435403205451 3866632807
6 13 1450005114454 3930544834
6 13 2135254510352 4953134588
6 13 2145555022413 5018649129
6 13 2500150125455 6170640875
6 15 133024510545125 124246559501
6 17 13435022253535055 4595333541803
6 17 15205355253553320 5341093125744
6 17 15205355253553321 5341093125745
6 18 105144341423554535 19418246235419
7 1 0 0
7 1 1 1
7 1 2 2
7 1 3 3
7 1 4 4
7 1 5 5
7 1 6 6
7 2 13 10
7 2 34 25
7 2 44 32
7 2 63 45
7 3 250 133
7 3 251 134
7 3 305 152
7 3 505 250
7 5 12205 3190
7 5 12252 3222
7 5 13350 3612
7 5 13351 3613
7 5 15124 4183
7 5 36034 9286
7 6 205145 35411
7 7 1424553 191334
7 7 1433554 193393
7 7 3126542 376889
7 7 4355653 535069
7 7 6515652 794376
7 9 125543055 8094840
7 9 161340144 10883814
7 9 254603255 16219922
7 9 336133614 20496270
7 9 542662326 32469576
7 9 565264226 34403018
7 11 13210652042 416002778
7 11 13213641035 416352977
7 11 13261421245 420197083
7 11 23662020022 725781499
7 11 52112660266 1500022495
7 13 1064120355440 15705029375
7 13 1064120355441 15705029376
7 13 2034134515166 28700208851
7 15 130101365302206 970930659537
7 15 130140113350606 972004335826
7 15 132336514346053 1003624386355
7 15 206161224560204 1443220146575
7 15 213452362516546 1504283967871
7 15 331634066630612 2352056093102
7 17 10531555446303336 36940082141157
7 17 13604563043402624 51612024946703
7 17 14010136562101652 52323166511954
7 17 30361566041660613 102340463411217
7 19 1064122116235164222 1847703627580701
7 19 1354465546206645132 2514834742553772
7 19 1626613640500155056 3123368686057682
7 21 144063133620460566064 132116164569671440
7 23 10063636240645406233515 3984625384955273973
7 23 10114351255652662661245 4008396591708493297
7 23 11406334566055365366500 4798127097158078159
7 23 11406334566055365366501 4798127097158078160
7 23 12616604301406016036306 5528252581301500133
8 1 0 0
8 1 1 1
8 1 2 2
8 1 3 3
8 1 4 4
8 1 5 5
8 1 6 6
8 1 7 7
8 2 24 20
8 2 64 52
8 3 134 92
8 3 205 133
8 3 463 307
8 3 660 432
8 3 661 433
8 5 40663 16819
8 5 42710 17864
8 5 42711 17865
8 5 60007 24583
8 5 62047 25639
8 6 636703 212419
8 7 3352072 906298
8 7 3352272 906426
8 7 3451473 938811
8 7 4217603 1122179
8 7 7755336 2087646
8 8 16450603 3821955
8 8 63717005 13606405
8 9 233173324 40695508
8 10 3115653067 423056951
8 10 4577203604 637339524
8 11 61777450236 6710775966
8 12 147402312024 13892162580
8 12 360507431167 32298119799
8 13 1323324472142 97095152738
8 13 1334013135714 98250308556
8 13 1334413135714 98317417420
8 13 1647540560022 125586038802
8 13 3017145740336 208198418654
8 13 4327761411137 303865139807
8 13 7173174033776 497577637886
8 16 1711433054170213 66627168170123
8 16 1711433054370212 66627168235658
8 18 216040024735257750 4998382669357032
8 18 216040024735257751 4998382669357033
8 18 223407227764431646 5190196317533094
8 19 1211032521007154743 22836489425312227
8 19 1232302004675154740 23446124043229664
8 19 1232302004675154741 23446124043229665
8 19 1232723462574551323 23464924977222355
8 19 1253541301300706674 24055210066546108
8 19 1253541301302706674 24055210067070396
8 19 3124771732711450547 57033449858814311
8 19 3124775743435202477 57033725899507007
8 20 11217560142462145753 167186253795085291
8 20 26270437173711052754 402810430791374316
8 21 275625505717602076277 3418993385062038719
8 24 174521742707223255307557 1149785975022171819887
8 25 1327077532453743552700004 6706538572304307945476
8 25 1561571732777266703512144 8132598905239700935780
8 25 1567777710567602656453055 8190352424223928636973
8 25 1567777750567602656453054 8190353550123835479596
8 29 11254613377540170731271074472 22576559626136290521807162
9 1 0 0
9 1 1 1
9 1 2 2
9 1 3 3
9 1 4 4
9 1 5 5
9 1 6 6
9 1 7 7
9 1 8 8
9 2 45 41
9 2 55 50
9 3 150 126
9 3 151 127
9 3 570 468
9 3 571 469
9 4 2446 1824
9 5 12036 8052
9 5 12336 8295
9 5 14462 9857
9 7 2225764 1198372
9 7 6275850 3357009
9 7 6275851 3357010
9 8 12742452 6287267
9 9 356614800 156608073
9 9 356614801 156608074
9 10 1033366170 403584750
9 10 1033366171 403584751
9 10 1455770342 586638974
9 10 8463825582 3302332571
9 12 131057577510 42256814922
9 12 131057577511 42256814923
9 12 358376613717 114842637961
9 12 486350373816 155896317510
9 13 1854014534430 552468844242
9 13 1854014534431 552468844243
9 13 2257238313123 647871937482
9 13 2376681144634 686031429775
9 13 2376684144633 686033024097
9 13 4255437336818 1212041747339
9 16 1501322621437658 320659684133768
9 17 14172228544021147 2717892501113815
9 17 25081352574053826 4756225997157666
9 17 35812034841853506 6774649666149786
9 18 223786486131001012 37860400025315399
9 19 1042226366438765012 157971147790033100
9 21 237082876620762854346 29433987159556011216
9 21 318880557521176087267 39172976026972151857
9 22 1537167772620206177887 175339949258468788828
9 23 17632082685752060101830 1828017958471661276001
9 23 17632082685752060101831 1828017958471661276002
9 23 17848085426122761171423 1854585265671831254178
9 25 1817532181082832263512440 152485824660312142632447
9 25 1817532181082832263512441 152485824660312142632448
9 25 1832330762825040066527288 153883752730136060093378
9 26 12438445561067680143612802 916767170213059160164793
9 26 21050515474420208822838438 1520549243223774953668208
9 29 11727236157010148446208821268 628740460999264727921582627
9 30 104836124432728001478001038310 4997059916255135935307115147
9 30 104836124432728001478001038311 4997059916255135935307115148
 

The next table lists base-10 PPDIs. The representations, in this table, are the normal decimal representations, so an additional column is unnecessary.
 
Order Number
1 0
1 1
1 2
1 3
1 4
1 5
1 6
1 7
1 8
1 9
3 153
3 370
3 371
3 407
4 1634
4 8208
4 9474
5 54748
5 92727
5 93084
6 548834
7 1741725
7 4210818
7 9800817
7 9926315
8 24678050
8 24678051
8 88593477
9 146511208
9 472335975
9 534494836
9 912985153
10 4679307774
11 32164049650
11 32164049651
11 40028394225
11 42678290603
11 44708635679
11 49388550606
11 82693916578
11 94204591914
14 28116440335967
16 4338281769391370
16 4338281769391371
17 21897142587612075
17 35641594208964132
17 35875699062250035
19 1517841543307505039
19 3289582984443187032
19 4498128791164624869
19 4929273885928088826
20 63105425988599693916
21 128468643043731391252
21 449177399146038697307
23 21887696841122916288858
23 27879694893054074471405
23 27907865009977052567814
23 28361281321319229463398
23 35452590104031691935943
24 174088005938065293023722
24 188451485447897896036875
24 239313664430041569350093
25 1550475334214501539088894
25 1553242162893771850669378
25 3706907995955475988644380
25 3706907995955475988644381
25 4422095118095899619457938
27 121204998563613372405438066
27 121270696006801314328439376
27 128851796696487777842012787
27 174650464499531377631639254
27 177265453171792792366489765
29 14607640612971980372614873089
29 19008174136254279995012734740
29 19008174136254279995012734741
29 23866716435523975980390369295
31 1145037275765491025924292050346
31 1927890457142960697580636236639
31 2309092682616190307509695338915
32 17333509997782249308725103962772
33 186709961001538790100634132976990
33 186709961001538790100634132976991
34 1122763285329372541592822900204593
35 12639369517103790328947807201478392
35 12679937780272278566303885594196922
37 1219167219625434121569735803609966019
38 12815792078366059955099770545296129367
39 115132219018763992565095597973971522400
39 115132219018763992565095597973971522401
 
Below are additional tables of PPDIs.

Complete List of Base-11 PPDIs (revised and corrected 5/31/2016)

Complete List of Base-12 PPDIs (revised and corrected 8/16/2016)

Dik Winter independently found PPDIs in bases 2 through 16. His listing for bases 2 through 12 agrees with my own. I have no reason to doubt the accuracy of his listing for bases 13 through 16. Note that Winter does not show 0 as a PPDI in any base. By virtue of the definitions noted earlier, however, 0 should be considered a PPDI in all bases, albeit a trivial one.

 

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