Table of Values of OEIS A158235, A158236, and A158237.
Michael De Vlieger, St. Louis, MO, 201709081130.

Extended using T. D. Noe's Mathematica program at A158235.

Table 1. Relation among OEIS A158235, A158236, and A158237.

n = index.
A158235 = number m such that m^2 is a repdigit d of L digits in base b.
A158236 = base b for which m^2 is an L-digit repetition of digit d.
A158237 = digit d that is repeated L times and represents m^2 in base b.
L = the number of digits d repeated L times to represent m^2 in base b.

Range covered: 2 <= m <= 101000, 2 <= b <= (m - 1).

  n        m           m^2      b      d   L
-------------------------------------------------
  1       11           121      3      1   5
  2       20           400      7      1   4
  3       39          1521     22      3   3
  4       40          1600      7      4   4
  5       49          2401     18      7   3
  6       78          6084     22     12   3
  7      133         17689     30     19   3
  8      247         61009     68     13   3
  9      494        244036     68     52   3
 10      543        294849    313      3   3
 11     1086       1179396    313     12   3
 12     1218       1483524     41     21   4
 13     1629       2653641    313     27   3
 14     1651       2725801    146    127   3
 15     1729       2989441    653      7   3
 16     2172       4717584    313     48   3
 17     2289       5239521    499     21   3
 18     2715       7371225    313     75   3
 19     3097       9591409    710     19   3
 20     3258      10614564    313    108   3
 21     3458      11957764    653     28   3
 22     3801      14447601    313    147   3
 23     4171      17397241    423     97   3
 24     4344      18870336    313    192   3
 25     4503      20277009    292    237   3
 26     4578      20958084    499     84   3
 27     4887      23882769    313    243   3
 28     5187      26904969    653     63   3
 29     5430      29484900    313    300   3
 30     6194      38365636    710     76   3
 31     6231      38825361    439    201   3
 32     6867      47155689    499    189   3
 33     6916      47831056    653    112   3
 34     7303      53333809    699    109   3
 35     7540      56851600     99     58   4
 36     7563      57198969   4366      3   3
 37     8342      69588964    423    388   3
 38     8645      74736025    653    175   3
 39     8773      76965529    521    283   3
 40     9139      83521321    581    247   3
 41     9156      83832336    499    336   3
 42     9291      86322681    710    171   3
 43    10374     107619876    653    252   3
 44    12103     146482609    653    343   3
 45    12388     153462544    710    304   3
 46    12439     154728721   4701      7   3
 47    13832     191324224    653    448   3
 48    14606     213335236    699    436   3
 49    15126     228795876   4366     12   3
 50    15485     239785225    710    475   3
 51    15561     242144721    653    567   3
 52    16627     276457129   4611     13   3
 53    16761     280931121    787    453   3
 54    17157     294362649   2272     57   3
 55    18582     345290724    710    684   3
 56    20280     411278400    239     30   4
 57    22021     484924441   2819     61   3
 58    22393     501446449   1047    457   3
 59    22689     514790721   4366     27   3
 60    24878     618914884   4701     28   3
 61    30252     915183504   4366     48   3
 62    33254    1105828516   4611     52   3
 63    34314    1177450596   2272    228   3
 64    37051    1372776601   4526     67   3
 65    37209    1384509681   1714    471   3
 66    37317    1392558489   4701     63   3
 67    37387    1397787769   1353    763   3
 68    37815    1429974225   4366     75   3
 69    40560    1645113600    239    120   4
 70    44042    1939697764   2819    244   3
 71    45201    2043130401   1660    741   3
 72    45378    2059162884   4366    108   3
 73    47257    2233224049   1425   1099   3
 74    49756    2475659536   4701    112   3
 75    49881    2488114161   4611    117   3
 76    51471    2649263841   2272    513   3
 77    51961    2699945521   2174    571   3
 78    52941    2802749481   4366    147   3
 79    60504    3660734016   4366    192   3
 80    62195    3868218025   4701    175   3
 81    65379    4274413641   6205    111   3
 82    66063    4364319969   2819    549   3
 83    66211    4383896521   2198    907   3
 84    66508    4423314064   4611    208   3
 85    68067    4633116489   4366    243   3
 86    68628    4709802384   2272    912   3
 87    69601    4844299201   2060   1141   3
 88    74102    5491106404   4526    268   3
 89    74634    5570233956   4701    252   3
 90    75630    5719896900   4366    300   3
 91    83135    6911428225   4611    325   3
 92    83193    6921075249   4366    363   3
 93    85785    7359066225   2272   1425   3
 94    87073    7581707329   4701    343   3
 95    88084    7758791056   2819    976   3
 96    90756    8236651536   4366    432   3
 97    98319    9666625761   4366    507   3
 98    99512    9902638144   4701    448   3
 99    99762    9952456644   4611    468   3
100   100191   10038236481   3019   1101   3

////// End Table 1 //////

Table 2. Bases b that appear in the range covered in the above table:

   b  ->  Numbers m for which m^2 is an L-digit repdigit k in base b
--------------------------------------------------------------------
   3  ->  1                                                 
   7  ->  2, 4                                              
  18  ->  5                                                 
  22  ->  3, 6                                              
  30  ->  7                                                 
  41  ->  12                                                
  68  ->  8, 9                                              
  99  ->  35                                                
 146  ->  14                                                
 239  ->  56, 69                                            
 292  ->  25                                                
 313  ->  10, 11, 13, 16, 18, 20, 22, 24, 27, 29            
 423  ->  23, 37                                            
 439  ->  31                                                
 499  ->  17, 26, 32, 41                                    
 521  ->  39                                                
 581  ->  40                                                
 653  ->  15, 21, 28, 33, 38, 43, 44, 47, 51                
 699  ->  34, 48                                            
 710  ->  19, 30, 42, 45, 50, 55                            
 787  ->  53                                                
1047  ->  58                                                
1353  ->  67                                                
1425  ->  73                                                
1660  ->  71                                                
1714  ->  65                                                
2060  ->  87                                                
2174  ->  77                                                
2198  ->  83                                                
2272  ->  54, 63, 76, 86, 93                                
2819  ->  57, 70, 82, 95                                    
3019  ->  100                                               
4366  ->  36, 49, 59, 61, 68, 72, 78, 79, 85, 90, 92, 96, 97
4526  ->  64, 88                                            
4611  ->  52, 62, 75, 84, 91, 99                            
4701  ->  46, 60, 66, 74, 80, 89, 94, 98                    
6205  ->  81                                                

////// End Table 2 //////

Table 3. Frequency k and Decomposition of Base b in A158236 for 2 <= m <= 101000.

k = frequency of b in A158236.
p = prime divisors of b.
q = prime divisors of b(b^2 - 1) = (b - 1) * b * (b + 1).

   b    k   p                   q
-----------------------------------------------------------------------
   3    1   3                   2, 2, 2, 3                            
   7    2   7                   2, 2, 2, 2, 3, 7                      
  18    1   2, 3, 3             2, 3, 3, 17, 19                       
  22    2   2, 11               2, 3, 7, 11, 23                       
  30    1   2, 3, 5             2, 3, 5, 29, 31                       
  41    1   41                  2, 2, 2, 2, 3, 5, 7, 41               
  68    2   2, 2, 17            2, 2, 3, 17, 23, 67                   
  99    1   3, 3, 11            2, 2, 2, 3, 3, 5, 5, 7, 7, 11         
 146    1   2, 73               2, 3, 5, 7, 7, 29, 73                 
 239    2   239                 2, 2, 2, 2, 2, 3, 5, 7, 17, 239       
 292    1   2, 2, 73            2, 2, 3, 73, 97, 293                  
 313   10   313                 2, 2, 2, 2, 3, 13, 157, 313           
 423    2   3, 3, 47            2, 2, 2, 2, 3, 3, 47, 53, 211         
 439    1   439                 2, 2, 2, 2, 3, 5, 11, 73, 439         
 499    4   499                 2, 2, 2, 3, 5, 5, 5, 83, 499          
 521    1   521                 2, 2, 2, 2, 3, 3, 5, 13, 29, 521      
 581    1   7, 83               2, 2, 2, 3, 5, 7, 29, 83, 97          
 653    9   653                 2, 2, 2, 3, 109, 163, 653             
 699    2   3, 233              2, 2, 2, 3, 5, 5, 7, 233, 349         
 710    6   2, 5, 71            2, 3, 3, 5, 71, 79, 709               
 787    1   787                 2, 2, 2, 3, 131, 197, 787             
1047    1   3, 349              2, 2, 2, 2, 3, 131, 349, 523          
1353    1   3, 11, 41           2, 2, 2, 2, 3, 11, 13, 13, 41, 677    
1425    1   3, 5, 5, 19         2, 2, 2, 2, 2, 3, 5, 5, 19, 23, 31, 89
1660    1   2, 2, 5, 83         2, 2, 3, 5, 7, 11, 79, 83, 151        
1714    1   2, 857              2, 3, 5, 7, 7, 7, 571, 857            
2060    1   2, 2, 5, 103        2, 2, 3, 3, 5, 29, 71, 103, 229       
2174    1   2, 1087             2, 3, 5, 5, 29, 41, 53, 1087          
2198    1   2, 7, 157           2, 3, 7, 13, 13, 13, 157, 733         
2272    5   2, 2, 2, 2, 2, 71   2, 2, 2, 2, 2, 3, 71, 757, 2273       
2819    4   2819                2, 2, 2, 3, 5, 47, 1409, 2819         
3019    1   3019                2, 2, 2, 3, 5, 151, 503, 3019         
4366   13   2, 37, 59           2, 3, 3, 5, 11, 37, 59, 97, 397       
4526    2   2, 31, 73           2, 3, 3, 5, 5, 31, 73, 181, 503       
4611    6   3, 29, 53           2, 2, 2, 3, 5, 29, 53, 461, 1153      
4701    8   3, 1567             2, 2, 2, 3, 5, 5, 47, 1567, 2351      
6205    1   5, 17, 73           2, 2, 2, 3, 5, 11, 17, 29, 47, 73, 107

////// End Table 3 //////