HOME plateWON | World!OfNumbers   Numbers whose digits occurwith same frequency ~ Classification P3 Class_P2 Class_P4 Class_P5 Classification P3 Frequency with which the digits occur in P3
'd' differ.
digits
1234567...
13000000...
22100000...
34110000...
4710000?...
5835124?...
61081945???...
7328110????...
8441831????...
90746858????...
100138?????...
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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

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Contributions

Jeff Heleen (email) from New Hampshire, USA.

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