In prime bases, k=b. In others, k appears to be divisible by b-1. Conjecture: If the prime factorization of composite b is p1^e1 * p2^e2 * p3^e3 * ..., then the factorization of k/(b-1), and x also, are each a product of powers of primes that are a subset of {p1, p2, p3, ...}.
I'll make this somewhat stronger: If the prime factorization of composite b is p_1^e_1 * p_2^e_2 * ... * p_n^e_n, then the factorization of x*k/(b-1) is p_1^f_1 * p_2^f_2 * ... * p_n^f_n. If e_1=e_2=...=e_n, then f_1=f_2=...=f_n. Since Tom Rokicki had multiple x's for some bases in his list < http://tomas.rokicki.com/k.txt >, I should clarify that I take x to be the *smallest* number that yields k. Special case 1: b = p^e, e>1 -> x = p and k/(b-1) = p^(e-1) Special case 2: b = p*q, q>p -> x = q^ceil(log_p(q)) and k/(b-1) = p^ceil(log_p(q)) (This is just a guessed extension of Tom Rokicki's formulation provided on December 16.) I've been looking at b = p*q*r, r>q>p. There are two difficulties: What is the exponent (f) of x*k/(b-1) and how does one determine the separation of x*k/(b-1) into its components x and k/(b-1)? base b-factored x*k/(b-1) x k/(b-1) 030 2*3*5 2^3*3^3*5^3 2^3*5^3 3^3 042 2*3*7 2^2*3^2*7^2 7^2 2^2*3^2 066 2*3*11 2^6*3^6*11^6 3^6*11^6 2^6 070 2*5*7 2^6*5^6*7^6 5^6*7^6 2^6 078 2*3*13 2^3*3^3*13^3 3*13^3 2^3*3^2 102 2*3*17 2^5*3^5*17^5 3^4*17^5 2^5*3 105 3*5*7 3^4*5^4*7^4 5^4*7^4 3^4 110 2*5*11 2^2*5^2*11^2 11^2 2^2*5^2 114 2*3*19 2^3*3^3*19^3 2*19^3 2^2*3^3 130 2*5*13 2^7*5^7*13^7 5^7*13^7 2^7 138 2*3*23 2^7*3^7*23^7 3^7*23^7 2^7 154 2*7*11 2^7*7^7*11^7 7^7*11^7 2^7 165 3*5*11 3^3*5^3*11^3 5^2*11^3 3^3*5 170 2*5*17 2^5*5^5*17^5 5^4*17^5 2^5*5 174 2*3*29 2^4*3^4*29^4 2^3*29^4 2*3^4 182 2*7*13 2^2*7^2*13^2 2^2*7^2 13^2 186 2*3*31 2^4*3^4*31^4 2^3*31^4 2*3^4 190 2*5*19 2^5*5^5*19^5 5^4*19^5 2^5*5 195 3*5*13 3^2*5^2*13^2 3^2*5^2 13^2 222 2*3*37 2^3*3^3*37^3 37^3 2^3*3^3 230 2*5*23 2^3*5^3*23^3 5*23^3 2^3*5^2 231 3*7*11 3^3*7^3*11^3 7^2*11^3 3^3*7 238 2*7*17 2^5*7^5*17^5 7^4*17^5 2^5*7 246 2*3*41 2^5*3^5*41^5 2^5*41^5 3^5 255 3*5*17 3^5*5^5*17^5 5^5*17^5 3^5 258 2*3*43 2^8*3^8*43^8 3^8*43^8 2^8 266 2*7*19 2^8*7^8*19^8 7^8*19^8 2^8 273 3*7*13 3^5*7^5*13^5 7^5*13^5 3^5 282 2*3*47 2^8*3^8*47^8 3^8*47^8 2^8 285 3*5*19 3^5*5^5*19^5 5^5*19^5 3^5 286 2*11*13 2^8*11^8*13^8 11^8*13^8 2^8 290 2*5*29 2^8*5^8*29^8 5^8*29^8 2^8 310 2*5*31 2^8*5^8*31^8 5^8*31^8 2^8 318 2*3*53 2^5*3^5*53^5 3^3*53^5 2^5*3^2 322 2*7*23 2^8*7^8*23^8 7^8*23^8 2^8 345 3*5*23 3^5*5^5*23^5 5^5*23^5 3^5 354 2*3*59 2^4*3^4*59^4 2^2*59^4 2^2*3^4 357 3*7*17 3^3*7^3*17^3 3^3*17^3 7^3 366 2*3*61 2^4*3^4*61^4 2^2*61^4 2^2*3^4 370 2*5*37 2^6*5^6*37^6 5^5*37^6 2^6*5 374 2*11*17 2^5*11^5*17^5 11^4*17^5 2^5*11 385 5*7*11 5^3*7^3*11^3 5^3*11^3 7^3 399 3*7*19 3^2*7^2*19^2 3^2*7^2 19^2 402 2*3*67 2^7*3^7*67^7 3^6*67^7 2^7*3 406 2*7*29 2^3*7^3*29^3 7*29^3 2^3*7^2 410 2*5*41 2^4*5^4*41^4 5^2*41^4 2^4*5^2 418 2*11*19 2^2*11^2*19^2 2^2*11^2 19^2 426 2*3*71 2^7*3^7*71^7 3^6*71^7 2^7*3 429 3*11*13 3^2*11^2*13^2 3*13^2 3*11^2 430 2*5*43 2^4*5^4*43^4 5^2*43^4 2^4*5^2 434 2*7*31 2^3*7^3*31^3 7*31^3 2^3*7^2 435 3*5*29 3^4*5^4*29^4 5^3*29^4 3^4*5 438 2*3*73 2^4*3^4*73^4 3*73^4 2^4*3^3 442 2*13*17 2^5*13^5*17^5 13^4*17^5 2^5*13 455 5*7*13 5^3*7^3*13^3 5^3*13^3 7^3 465 3*5*31 3^4*5^4*31^4 5^3*31^4 3^4*5 470 2*5*47 2^4*5^4*47^4 5^2*47^4 2^4*5^2 474 2*3*79 2^4*3^4*79^4 3*79^4 2^4*3^3 483 3*7*23 3^2*7^2*23^2 23^2 3^2*7^2 494 2*13*19 2^5*13^5*19^5 13^4*19^5 2^5*13 498 2*3*83 2^5*3^5*83^5 2^4*83^5 2*3^5 506 2*11*23 2^2*11^2*23^2 23^2 2^2*11^2 518 2*7*37 2^9*7^9*37^9 7^9*37^9 2^9 530 2*5*53 2^9*5^9*53^9 5^9*53^9 2^9 534 2*3*89 2^9*3^9*89^9 3^9*89^9 2^9