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Numbers whose digits occur
with same frequency ~ Classification P3
rood Class_P2 rood Class_P4 rood Class_P5


led Classification P3 led


 Frequency with which the digits occur in P3
'd' differ.
digits		1234567...
13000000...
22100000...
34110000...
4710000?...
5835124?...
61081945???...
7328110????...
8441831????...
90746858????...
100138?????...


The Making of an Exhaustive List

The smallest solutions

P3(1.1)A = 03 = 0
P3(1. >1)A = nihil

P3(2.1)A = 33 = 27
P3(2.2)A = 113 = 1331 = P3(2.2)Z
P3(2. >2)A = nonexistant ?

P3(3.1)A = 53 = 125
P3(3.2)A = 623 = 238328 = P3(3.2)Z
P3(3.3)A = 8883 = 700227072 = P3(3.3)Z
P3(3. >3)A = nonexistant ?

P3(4.1)A = 123 = 1728
P3(4.2)A = 3033 = 27818127 = P3(4.2)Z
P3(4. >2)A = nonexistant ?

P3(5.1)A = 223 = 10648
P3(5.2)A = 14123 = 2815166528
P3(5.3)A = 565923 = 181244621426688
P3(5.4)A = 28304793 = 22676697737363992239 = P3(5.4)Z
P3(5.5)A = 1200231423 = 1728999927211172788179288
P3(5.6)A = 63517831053 = 256263633328535368685258882625

P3(6.1)A = 593 = 205379
P3(6.2)A = 47553 = 107510668875
P3(6.3)A = 4969413 = 122719757596965621
P3(6.4)A = 478021093 = 109229808820515515981029
P3(6.5)A = 48497955713 = 114069699564596150060045954411

P3(7.1)A = 1353 = 2460375
P3(7.2)A = 221113 = 10809986553631
P3(7.3)A = 46541913 = 100816729790622689871
P3(7.4)A = 10018027813 = 1005418098917074987897545541

P3(8.1)A = 2893 = 24137569
P3(8.2)A = 1101853 = 1337726800581625
P3(8.3)A = 464405463 = 100159454656890483891336
P3(8.4)A = 215469218163 = 10003585938508415641831399466496

P3(9.1)A = nihil
P3(9.2)A = 4700953 = 103885969226107375
P3(9.3)A = 4642096173 = 100032794277593492545888113
P3(9.4)A = 4641655790223 = 100004327671547355628148253326718648

P3(10.1)A = nihil
P3(10.2)A = 21584793 = 10056421854778936239
P3(10.3)A = 46421105943 = 100033726751278963952981464584

The largest solutions

P3(1.1)Z = 23 = 8
P3(1. >1)Z = nihil

P3(2.1)Z = 43 = 64
P3(2.2)Z = 113 = 1331 = P3(2.2)A
P3(2. >2)Z = nihil ?

P3(3.1)Z = 93 = 729
P3(3.2)Z = 623 = 238328 = P3(3.2)A
P3(3.3)Z = 8883 = 700227072 = P3(3.3)A
P3(3. >3)Z = nonexistant ?

P3(4.1)Z = 213 = 9261
P3(4.2)Z = 3033 = 27818127 = P3(4.2)A
P3(4. >2)Z = nonexistant ?

P3(5.1)Z = 413 = 68921
P3(5.2)Z = 20783 = 8972978552
P3(5.3)Z = 781833 = 477899958554487
P3(5.4)Z = 28304793 = 22676697737363992239 = P3(5.4)A
P3(5.5)Z = 1311588253 = 2256277705056006702765625
P3(5.6)Z = 84396494823 = 601136681133830360316800808168

P3(6.1)Z = 973 = 912673
P3(6.2)Z = 97093 = 915215787829
P3(6.3)Z = 9884583 = 965772115692567912
P3(6.4)Z = 992599963 = 977963756545433564479936
P3(6.5)Z = 99467068043 = 984097094780740676997868806464

P3(7.1)Z = 2033 = 8365427
P3(7.2)Z = 456753 = 95287441921875
P3(7.3)Z = 99649323 = 989516449813334165568
P3(7.4)Z = 21527358243 = 9976362336324497457957556224

P3(8.1)Z = 3193 = 32461759
P3(8.2)Z = 2109163 = 9382716173855296
P3(8.3)Z = 997853643 = 993574730695766088308544
P3(8.4)Z = 464150896823 = 99994838144343656268511251322568

P3(9.1)Z = nihil
P3(9.2)Z = 9826173 = 948752253468679113
P3(9.3)Z = 9997745733 = 999323871440541388225070517
P3(9.4)Z = 9999896039123 = 999968812060234813516120343055446528

P3(10.1)Z = nihil
P3(10.2)Z = 46317933 = 99368200745116834257
P3(10.3)Z = 99992577813 = 999777350826262438514680310541

The total number of solutions

P3(1.1)T = 3
03 = 0
13 = 1
23 = 8
P3(1. >1)T = 0    all nonexistant ?

P3(2.1)T = 2
33 = 27
43 = 64
P3(2.2)T = 1    a unique solution !
113 = 1331
P3(2. >2)T = 0    all nonexistant ?

P3(3.1)T = 4
53 = 125
63 = 216
83 = 512
93 = 729
P3(3.2)T = 1    a unique solution !
623 = 238328
P3(3.3)T = 1    a unique solution !
8883 = 700227072
P3(3. >3)T = 0    all nonexistant ?

P3(4.1)T = 7
123 = 1728  |  183 = 5832
133 = 2197  |  193 = 6859
163 = 4096  |  213 = 9261
173 = 4913
P3(4.2)T = 1    a unique solution !
3033 = 27818127
P3(4. >2)T = 0    all nonexistant ?

P3(5.1)T = 8
223 = 10648  |  323 = 32768
243 = 13824  |  353 = 42875
273 = 19683  |  383 = 54872
293 = 24389  |  413 = 68921
P3(5.2)T = 3
14123 = 2815166528
16943 = 4861163384
20783 = 8972978552
P3(5.3)T = 5
565923 = 181244621426688
585243 = 200448128101824
655773 = 282003588255033
708693 = 355933540044909
781833 = 477899958554487
P3(5.4)T = 1    a unique solution !
28304793 = 22676697737363992239
P3(5.5)T = 2
1200231423 = 1728999927211172788179288
1311588253 = 2256277705056006702765625
P3(5.6)T = 4
63517831053 = 256263633328535368685258882625
73641800553 = 399367937595765965775933666375
80011000803 = 512211244405555444021120512000
84396494823 = 601136681133830360316800808168

P3(6.1)T = 10
593 = 205379  |  763 = 438976
663 = 287496  |  843 = 592704
693 = 328509  |  883 = 681472
733 = 389017  |  933 = 804357
753 = 421875  |  973 = 912673
P3(6.2)T = 8
47553 = 107510668875  |  64433 = 267463420307
53993 = 157376536199  |  90783 = 748118742552
61813 = 236143627741  |  94133 = 834034807997
62743 = 246963938824  |  97093 = 915215787829
P3(6.3)T = 19
P3(6.4)T = 45

P3(7.1)T = 3
1353 = 2460375
1453 = 3048625
2033 = 8365427
P3(7.2)T = 28
P3(7.3)T = 110

P3(8.1)T = 4
2893 = 24137569
2973 = 26198073
3023 = 27543608
3193 = 32461759
P3(8.2)T = 41
P3(8.3)T = 831

P3(9.1)T = 0
P3(9.2)T = 74
P3(9.3)T = 6858

P3(10.1)T = 0
P3(10.2)T = 138
P3(10.3)T = ?

Here is my collection of such numbers with palindromic cuberoots.
03 = 0
13 = 1
23 = 8


Contributions

Jeff Heleen (email) from New Hampshire, USA.
 

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