HOME plateWON | World!OfNumbers Numbers whose digits occurwith same frequency ~ Classification P5 Class_P2 Class_P3 Class_P4

Classification P5

Frequency with which the digits occur in P5
'd' differ.
digits
1234567...
12000000...
21000000...
31000000...
42000000...
52000000...
61000000...
7100000?...
81030111?...
9011884??...
10007133???...

The Making of an Exhaustive List

The smallest solutions

P5(1.1)A = 05 = 0
P5(1. >1)A = nihil

P5(2.1)A = 25 = 32 = P5(2.1)Z
P5(2. >1)A = nihil

P5(3.1)A = 35 = 243 = P5(3.1)Z
P5(3. >1)A = nihil

P5(4.1)A = 45 = 1024
P5(4. >1)A = nihil

P5(5.1)A = 75 = 16807
P5(5. >1)A = nihil

P5(6.1)A = 145 = 537824 = P5(6.1)Z
P5(6. >1)A = nihil

P5(7.1)A = 165 = 1048576 = P5(7.1)Z
P5(7. >1)A = nihil

P5(8.1)A = 385 = 79235168 = P5(8.1)Z
P5(8.2)A = nihil
P5(8.3)A = 499955 = 312343781246875156246875
P5(8.4)A = nihil
P5(8.5)A = 754183845 = 2439979134100773706931016420916722663424 = P5(8.5)Z
P5(8.6)A = 25928774105 = 117195225794292252449115584887987847895470100000
P5(8.7)A = ?5 = ?

P5(9.1)A = nihil
P5(9.2)A = 29555 = 225313610074846875 = P5(9.2)Z
P5(9.3)A = 1931255 = 268653488211536407470703125 = P5(9.3)Z
P5(9.4)A = 103655895 = 119665765800843104737370354851986949
P5(9.5)A = 6315641855 = 100481814610246818738257752346024507337365625
P5(9.6)A = ?5 = ?

P5(10.1)A = nihil
P5(10.2)A = nihil
P5(10.3)A = 6439055 = 110690152879433875483274690625
P5(10.4)A = 631740395 = 1006220638584953725761454302767138894199

The largest solutions

P5(1.1)Z = 15 = 1
P5(1. >1)Z = nihil

P5(2.1)Z = 25 = 32 = P5(2.1)A
P5(2. >1)Z = nihil

P5(3.1)Z = 35 = 243 = P5(3.1)A
P5(3. >1)Z = nihil

P5(4.1)Z = 55 = 3125
P5(4. >1)Z = nihil

P5(5.1)Z = 85 = 32768
P5(5. >1)Z = nihil

P5(6.1)Z = 145 = 537824 = P5(6.1)A
P5(6. >1)Z = nihil

P5(7.1)Z = 165 = 1048576 = P5(7.1)A
P5(7. >1)Z = nihil

P5(8.1)Z = 385 = 79235168 = P5(8.1)A
P5(8.2)Z = nihil
P5(8.3)Z = 615575 = 883867634090375064499557
P5(8.4)Z = nihil
P5(8.5)Z = 754183845 = 2439979134100773706931016420916722663424 = P5(8.5)A
P5(8.6)Z = 38350203035 = 829541919891852538715325897428343419274377521743
P5(8.7)Z = ?5 = ?

P5(9.1)Z = nihil
P5(9.2)Z = 29555 = 225313610074846875 = P5(9.2)A
P5(9.3)Z = 1931255 = 268653488211536407470703125 = P5(9.3)A
P5(9.4)Z = 147688685 = 702645136480123183310487625524077568
P5(9.5)Z = 9927587155 = 964314153757084568107154987089060693334196875
P5(9.6)Z = ?5 = ?

P5(10.1)Z = nihil
P5(10.2)Z = nihil
P5(10.3)Z = 9849275 = 926872965448613570921013853407
P5(10.4)Z = 999008285 = 9950512253367684107778194042198520634368

The total number of solutions

P5(1.1)T = 2
05 = 0
15 = 1
P5(1. >1)T = 0

P5(2.1)T = 1    a unique solution !
25 = 32
P5(2. >1)T = 0    all nonexistant ?

P5(3.1)T = 1    a unique solution !
35 = 243
P5(3. >1)T = 0    all nonexistant ?

P5(4.1)T = 2
45 = 1024
55 = 3125
P5(4. >1)T = 0    all nonexistant ?

P5(5.1)T = 2
75 = 16807
85 = 32768
P5(5. >1)T = 0    all nonexistant ?

P5(6.1)T = 1    a unique solution !
145 = 537824
P5(6. >1)T = 0    all nonexistant ?

P5(7.1)T = 1    a unique solution !
165 = 1048576
P5(7. >1)T = 0    all nonexistant ?

P5(8.1)T = 1    a unique solution !
385 = 79235168
P5(8.2)T = 0
P5(8.3)T = 3
499955 = 312343781246875156246875
576155 = 634859804603764980759375
615575 = 883867634090375064499557
P5(8.4)T = 0
P5(8.5)T = 1    a unique solution !
754183845 = 2439979134100773706931016420916722663424
P5(8.6)T = 11
25928774105 = 117195225794292252449115584887987847895470100000
29374119185 = 218687652680475264071477052082441458720015161568
30543510065 = 265824327313481538631122845681762715546354487776
31455044795 = 307929672182062108611086171702308879332696738399
32258524895 = 349319251551263912652396312024040050366594601449
33832318565 = 443260270395509692467230060455427333476529995776
34053623795 = 457948529798626755266418971251122416415746887899
35743812575 = 583450975970633003445766344389490869567868798057
36588414425 = 655719271130757967396019103309526657120920563232
38235547895 = 817215487524126241979815788947692466565186572949
38350203035 = 829541919891852538715325897428343419274377521743
P5(8.7)T = ?

P5(9.1)T = 0
P5(9.2)T = 1    a unique solution !
29555 = 225313610074846875
P5(9.3)T = 1    a unique solution !
1931255 = 268653488211536407470703125
P5(9.4)T = 8
103655895 = 119665765800843104737370354851986949
112281035 = 178456322746061358045561821732080743
120368285 = 252673835969632125494758974118714368
123437885 = 286577619853439827259931426441735168
126083025 = 318627322049731980106674841799864032
127360245 = 335096077304295591976171102543642624
141754055 = 572370680423381730540165646748128125
147688685 = 702645136480123183310487625524077568
P5(9.5)T = 84
P5(9.6)T = ?

P5(10.1)T = 0
P5(10.2)T = 0
P5(10.3)T = 7
6439055 = 110690152879433875483274690625
6800615 = 145458581697864327220703996301
7205585 = 194242706843325709850513196768
7751135 = 279785436151445000683198226793
8405015 = 419460598737334268928156702501
8786135 = 523586118778694132765090044293
9849275 = 926872965448613570921013853407
P5(10.4)T = 133
P5(10.5)T = ?

Here is my collection of such numbers with palindromic fifth roots.
05 = 0
15 = 1

Contributions

Jeff Heleen (email) from New Hampshire, USA.