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 //////