HOME plateWON | World!OfNumbers Numbers whose digits occurwith same frequency ~ Classification P4 Class_P2 Class_P3 Class_P5

Classification P4

Frequency with which the digits occur in P4
'd' differ.
digits
1234567...
12000000...
22000000...
32000000...
43000000...
54100000...
6300000?...
751152??...
803540???...
90225759???...
1004172????...
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The Making of an Exhaustive List

The smallest solutions

P4(1.1)A = 04 = 0
P4(1. >1)A = nihil

P4(2.1)A = 24 = 16
P4(2. >1)A = nihil

P4(3.1)A = 44 = 256
P4(3. >1)A = nihil

P4(4.1)A = 64 = 1296
P4(4. >1)A = nihil

P4(5.1)A = 124 = 20736
P4(5.2)A = 2074 = 1836036801 = P4(5.2)Z
P4(5. >2)A = nihil

P4(6.1)A = 184 = 104976
P4(6. >1)A = nihil

P4(7.1)A = 324 = 1048576
P4(7.2)A = 24524 = 36147799388416 = P4(7.2)Z
P4(7.3)A = 1303984 = 289123718973983667216 = P4(7.3)Z
P4(7.4)A = 56942074 = 1051315345334684056886604801
P4(7.5)A = 4264248284 = 33065106952901022329359695121613056
P4(7.6)A = 183046410244 = 112265125207703310302573655130362165067776

P4(8.1)A = nihil
P4(8.2)A = 86924 = 5707933051146496
P4(8.3)A = 5690584 = 104863930698324110228496
P4(8.4)A = 570813914 = 10616422434730826078387783204161
P4(8.5)A = 56295463394 = 1004369679090466326141409237273977132241
P4(8.6)A = 5623795961074 = 100027225332362313560971766691937509957725591601

P4(9.1)A = nihil
P4(9.2)A = 242674 = 346788239145769521
P4(9.3)A = 31820434 = 102523677648570833624514801
P4(9.4)A = 5626062914 = 100188606547504649933395818497375761
P4(9.5)A = 1000052445224 = 100020979738358361519703325186685226719978256

P4(10.1)A = nihil
P4(10.2)A = 696364 = 23514473895962870016
P4(10.3)A = 178247194 = 100946384385027947782661539521
P4(10.4)A = 56237206624 = 1000218682348975505736229176987914443536

The largest solutions

P4(1.1)Z = 14 = 1
P4(1. >1)Z = nihil

P4(2.1)Z = 34 = 81
P4(2. >1)Z = nihil

P4(3.1)Z = 54 = 625
P4(3. >1)Z = nihil

P4(4.1)Z = 84 = 4096
P4(4. >1)Z = nihil

P4(5.1)Z = 174 = 83521
P4(5.2)Z = 2074 = 1836036801 = P4(5.2)A
P4(5. >2)Z = nihil

P4(6.1)Z = 254 = 390625
P4(6. >1)Z = nihil

P4(7.1)Z = 494 = 5764801
P4(7.2)Z = 24524 = 36147799388416 = P4(7.2)A
P4(7.3)Z = 1303984 = 289123718973983667216 = P4(7.3)A
P4(7.4)Z = 96816184 = 8786011595668091975085109776
P4(7.5)Z = 4873315404 = 56402464255939342395392324966560000
P4(7.6)Z = 311591389834 = 942631163354926125294614695396339214455521

P4(8.1)Z = nihil
P4(8.2)Z = 95954 = 8475784699200625
P4(8.3)Z = 9778474 = 914289286476569154757281
P4(8.4)Z = 992662854 = 97097282557302398839867366500625
P4(8.5)Z = 99980706944 = 9992285009045746654672422078758808756496
P4(8.6)Z = 9998175029464 = 999270211590737036231702317961557666152302935056

P4(9.1)Z = nihil
P4(9.2)Z = 300594 = 816390822057597361
P4(9.3)Z = 55774084 = 967675311854837821442359296
P4(9.4)Z = 9995924274 = 998370704423712389607141298663608241
P4(9.5)Z = 1778229245994 = 999887167503511209336573208035869782275261601

P4(10.1)Z = nihil
P4(10.2)Z = 808924 = 42817597245013360896
P4(10.3)Z = 316070524 = 998012460822049787453371535616
P4(10.4)Z = 99991129264 = 9996452176112247953600451338048028738576

The total number of solutions

P4(1.1)T = 2
04 = 0
14 = 1
P4(1. >1)T = 0

P4(2.1)T = 2
24 = 16
34 = 81
P4(2. >1)T = 0

P4(3.1)T = 2
44 = 256
54 = 625
P4(3. >1)T = 0

P4(4.1)T = 3
64 = 1296
74 = 2401
84 = 4096
P4(4. >1)T = 0

P4(5.1)T = 4
124 = 20736
134 = 28561
144 = 38416
174 = 83521
P4(5.2)T = 1    a unique solution !
2074 = 1836036801
P4(5. >2)T = 0

P4(6.1)T = 3
184 = 104976
234 = 279841
254 = 390625
P4(6. >1)T = 0

P4(7.1)T = 5
324 = 1048576
384 = 2085136
444 = 3748096
484 = 5308416
494 = 5764801
P4(7.2)T = 1    a unique solution !
24524 = 36147799388416
P4(7.3)T = 1    a unique solution !
1303984 = 289123718973983667216
P4(7.4)T = 5
56942074 = 1051315345334684056886604801
74237044 = 3037264324502656425730707456
82551384 = 4644054960561353915900113936
94543924 = 7989772631868111973223632896
96816184 = 8786011595668091975085109776
P4(7.5)T = 2
4264248284 = 33065106952901022329359695121613056
4873315404 = 56402464255939342395392324966560000

P4(8.1)T = 0
P4(8.2)T = 3
86924 = 5707933051146496
87214 = 5784490950217281
95954 = 8475784699200625
P4(8.3)T = 5
5690584 = 104863930698324110228496
6747784 = 207321173700596599312656
7706374 = 352695100362907795721361
9776434 = 913526563244942905031601
9778474 = 914289286476569154757281
P4(8.4)T = 40
P4(8.5)T = ?

P4(9.1)T = 0
P4(9.2)T = 2
242674 = 346788239145769521
300594 = 816390822057597361
P4(9.3)T = 25
P4(9.4)T = 759
P4(9.5)T = ?

P4(10.1)T = 0
P4(10.2)T = 4
696364 = 23514473895962870016
702154 = 24306341799881750625
770584 = 35259076387041812496
808924 = 42817597245013360896
P4(10.3)T = 172
P4(10.4)T = ?

Here is my collection of such numbers with palindromic fourth roots.
04 = 0
14 = 1

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Contributions

Jeff Heleen (email) from New Hampshire, USA.

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( © All rights reserved ) - Last modified : December 13, 2022.

Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com