Four weeks ago I discovered what I now know to be the smallest base-ten 3-balanced factorization integer. By the end of October I had completed my base-ten k-balanced list to 10^12 and decided to extend the search to 1.5 * 10^12, thus including (when finished) the first of the four 13-digit 2-balanced factorization integers. I was of course also hoping for more (small) 3-balanced examples. Yesterday and today one of my processors found these (which thusly become the second-, third-, and fourth-smallest):
1045675984884 = 2^2 * 3 * 17 * 67 * 103 * 359 * 2069
1046959786860 = 2^2 * 3 * 5 * 13 * 43 * 2087 * 14957
1047697856460 = 2^2 * 3^2 * 5 * 109 * 347 * 153889
My ill-fated, eight digits reverse search is now back on track, though I realize that it will take a good while to complete. It will rediscover the twenty-one 12-digit 2-balanced factorization integers and — more excitingly — has already come up with a k-balanced solution for k > 2.