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????...

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

Contributions

Jeff Heleen (email) from New Hampshire, USA.