Patterns| Base-10 integers in pair bases b1 and b2

 
 

Other Bases 1

 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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5 Responses to Patterns| Base-10 integers in pair bases b1 and b2

  1. Paul says:

    In terms of the specific pattern where we have two consecutive sets of 2 numbers and the 5th number is the last number plus 5 then those are the only ones I can find with bases from 2 to 21 and the integer up to 10 million. Also there are no more number with those 2 bases up to 100 million. Now if we relax the pattern match a little such that the match is a single number we find quite a few more, in the same range b1<20 && b2<21 and N <=10000000 there are just over 1000 examples, Do you want them all or just a few?

    Paul.

  2. Paul says:

    ok here’s a snapshot
    N b1 b2 b-1 b2
    2180306 , 5 , 8 , 1024232211 , 10242322
    2196768 , 5 , 8 , 1030244033 , 10302440
    2196996 , 5 , 8 , 1030300441 , 10303004
    2197000 , 5 , 8 , 1030301000 , 10303010
    2364049 , 5 , 8 , 1101122144 , 11011221
    2364050 , 5 , 8 , 1101122200 , 11011222
    4561049 , 5 , 8 , 2131423144 , 21314231
    4561050 , 5 , 8 , 2131423200 , 21314232
    4562971 , 5 , 8 , 2132003341 , 21320033
    4728099 , 5 , 8 , 2202244344 , 22022443
    4728100 , 5 , 8 , 2202244400 , 22022444
    4728331 , 5 , 8 , 2202301311 , 22023013
    6927049 , 5 , 8 , 3233131144 , 32331311
    6927050 , 5 , 8 , 3233131200 , 32331312
    7092384 , 5 , 8 , 3303424014 , 33034240
    7094306 , 5 , 8 , 3304004211 , 33040042
    7110793 , 5 , 8 , 3310021133 , 33100211
    9291331 , 5 , 8 , 4334310311 , 43343103
    9307793 , 5 , 8 , 4340322133 , 43403221
    9474843 , 5 , 8 , 4411143333 , 44111433
    9475074 , 5 , 8 , 4411200244 , 44112002
    9475075 , 5 , 8 , 4411200300 , 44112003

    1343 , 6 , 11 , 10115 , 1011
    1344 , 6 , 11 , 10120 , 1012
    2687 , 6 , 11 , 20235 , 2023
    2688 , 6 , 11 , 20240 , 2024
    4031 , 6 , 11 , 30355 , 3035
    4037 , 6 , 11 , 30405 , 3040
    4038 , 6 , 11 , 30410 , 3041
    4115 , 6 , 11 , 31015 , 3101
    4116 , 6 , 11 , 31020 , 3102
    5381 , 6 , 11 , 40525 , 4052
    5382 , 6 , 11 , 40530 , 4053
    5459 , 6 , 11 , 41135 , 4113
    5460 , 6 , 11 , 41140 , 4114
    6803 , 6 , 11 , 51255 , 5125
    6809 , 6 , 11 , 51305 , 5130
    6810 , 6 , 11 , 51310 , 5131
    1806335 , 6 , 11 , 102414355 , 1024143
    1806336 , 6 , 11 , 102414400 , 1024144
    3620730 , 6 , 11 , 205334350 , 2053343
    3725506 , 6 , 11 , 211503414 , 2115034
    3733503 , 6 , 11 , 212004423 , 2120044
    5539901 , 6 , 11 , 314423405 , 3144234
    5643982 , 6 , 11 , 320545314 , 3205453
    5652035 , 6 , 11 , 321050455 , 3210504
    5652036 , 6 , 11 , 321050500 , 3210505
    7459071 , 6 , 11 , 423512423 , 4235124
    7467068 , 6 , 11 , 424013432 , 4240134
    7571149 , 6 , 11 , 430135341 , 4301353
    9377547 , 6 , 11 , 532554323 , 5325543
    9385544 , 6 , 11 , 533055332 , 5330553
    9386238 , 6 , 11 , 533102450 , 5331024

    363 , 7 , 19 , 1026 , 102
    364 , 7 , 19 , 1030 , 103
    727 , 7 , 19 , 2056 , 205
    728 , 7 , 19 , 2060 , 206
    741 , 7 , 19 , 2106 , 210
    742 , 7 , 19 , 2110 , 211
    1105 , 7 , 19 , 3136 , 313
    1106 , 7 , 19 , 3140 , 314
    1469 , 7 , 19 , 4166 , 416
    1483 , 7 , 19 , 4216 , 421
    1484 , 7 , 19 , 4220 , 422
    1847 , 7 , 19 , 5246 , 524
    1848 , 7 , 19 , 5250 , 525
    1862 , 7 , 19 , 5300 , 530
    2225 , 7 , 19 , 6326 , 632
    2226 , 7 , 19 , 6330 , 633
    132508 , 7 , 19 , 1061215 , 10612
    137579 , 7 , 19 , 1112051 , 11120
    275159 , 7 , 19 , 2224133 , 22241
    412739 , 7 , 19 , 3336215 , 33362
    412984 , 7 , 19 , 3340015 , 33400
    550563 , 7 , 19 , 4452066 , 44520
    550564 , 7 , 19 , 4452100 , 44521
    555623 , 7 , 19 , 4502615 , 45026
    688143 , 7 , 19 , 5564151 , 55641
    693202 , 7 , 19 , 5614666 , 56146
    693215 , 7 , 19 , 5615015 , 56150

    17 , 4 , 17 , 101 , 10
    34 , 4 , 17 , 202 , 20
    51 , 4 , 17 , 303 , 30
    52 , 4 , 17 , 310 , 31
    291 , 4 , 17 , 10203 , 102
    596 , 4 , 17 , 21110 , 211
    887 , 4 , 17 , 31313 , 313
    901 , 4 , 17 , 32011 , 320
    4965 , 4 , 17 , 1031211 , 1031
    5203 , 4 , 17 , 1101103 , 1101
    10168 , 4 , 17 , 2132320 , 2132
    10406 , 4 , 17 , 2202212 , 2202
    15609 , 4 , 17 , 3303321 , 3303
    15623 , 4 , 17 , 3310013 , 3310
    88759 , 4 , 17 , 111222313 , 11122
    177754 , 4 , 17 , 223121122 , 22312
    181800 , 4 , 17 , 230120220 , 23012
    1518153 , 4 , 17 , 11302221021 , 113022
    1586951 , 4 , 17 , 12003130013 , 120031
    3105339 , 4 , 17 , 23312020323 , 233120

    Paul.

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