Defenition: a_d(n) = maximum starting value of X such that repeated replacement of X with X-ceiling(X/d) requires n steps to reach 0. A279075..A279080 for d = 5..10. Lim_{n->inf} a_d(n)/(d/(d-1))^n is given by (d = 2..20): d = 2 (trivial; a_2(n) = 2^n-1; see also A000007) 1 d = 3 (see also A083286) 1. 6222705028 8476731595 6950982899 3241130661 0556231303 7432185443 3873784339 9972748447 6383616539 8332334110 0723017878 1602121683 4013034803 0983903156 1672393427 9298395577 5893508605 6253447165 6841496841 1589360985 d = 4 (near A104540 but not the same) 2. 0763957307 9966671381 4071765362 7546067142 5179147509 8431738288 3037445073 6650918844 5100769774 9957129379 5269973777 8027330524 2072176927 0312850010 6722922572 7618565476 8477932585 8106049906 0644596892 0570401163 d = 5 (see also A279075) 2. 6872305827 0145442816 3834765673 3195732919 9286146873 4923248828 0899358628 5315257041 4116401354 9317852867 3268513218 9942977602 9347116672 1650861795 3087245976 4244678772 6494228923 1853583940 3080183605 0296611186 d = 6 (see also A279076) 3. 2438724975 1177521384 7348539055 1780261817 1089570674 9679701890 8045888646 0917601212 5147521051 5265397357 8028697136 9116136075 9087069704 3478227664 5962042330 0893427736 4773074574 1929289851 6361756379 4052493803 d = 7 (see also A279077) 4. 0371021121 5303193642 7914581111 9692295055 1168987041 9393981530 3926822181 8730121538 8442000508 9156833807 4279500836 5322868077 0002634723 4670204965 7931341224 3877880844 8778626688 1002998496 3125981949 2037066593 d = 8 (see also A279078) 4. 4221034795 9393228709 6044124458 0220122090 7917744900 6136309198 5598991291 6090477834 1160237716 1279733156 4052950730 4377764488 4545773202 6570364106 6522283180 6381520848 0594152739 5839954376 3820488172 6282062800 d = 9 (see also A279079) 5. 1954489639 2362185906 4609155721 9516994503 9729234281 0535477408 8156826138 3301754492 0961461782 3602377538 1012505190 9006060095 7341007724 6821831648 3983628666 2233072157 0733689839 5220348516 0054843512 5416324766 d = 10 (see also A279080) 5. 6065560113 6196116133 0578762946 8780726503 5051745268 9718054432 9304472812 5555331293 8659586107 0793987452 2230396780 3156159780 5759325687 6778336060 9645909056 8438856402 4788824287 1269107164 6642102144 3961256803 d = 11 6. 1922534221 7353887988 3192093422 5784345691 3454962934 2593615921 0944686097 2960766085 6555764276 3828470865 7165603223 5836422908 8831785968 1966275133 9012656564 5976410624 7003734706 1740937791 7140889426 6094585015 d = 12 6. 6336256864 3144123030 4390394531 7786586985 6820896435 8268381492 7364344696 6807376066 5000943782 0228281172 0371476047 5442213925 4161689987 2019472253 5596803676 8445647588 4823493965 0062325943 7664205670 1756248536 d = 13 7. 3765841748 8690604689 7312743568 0958081370 3320434355 1170677685 3050046804 5477033123 3735664836 2758600483 1472802137 0545616780 5747333108 5035082860 0104509462 3178314546 6588907175 0490352275 5811460094 7934516406 d = 14 7. 9049091486 3729643499 6543582680 9134191192 2600890011 4700501065 6717565862 0471112733 5694046826 6026946517 2701870535 9174390515 0156470082 7738971686 0132518544 0684460747 0140970605 3260851198 1768370483 7624242657 d = 15 8. 2999398470 9301260134 6489271155 4561531175 0972302946 3585762191 6697103973 1279486239 2501728554 1476757655 3954956285 6916685111 4548813734 7174732376 6177645181 3958240398 0918144185 2901496811 7089639642 8812742277 d = 16 8. 7693598707 1073580088 7050473639 2152595930 1296384250 0382136826 7265179601 1093358955 8713582297 2024078463 4752026649 8226231367 4917509543 7165540194 5172414827 5559804481 4462884272 9494048193 0927054587 8052726412 d = 17 9. 5738276555 8724583877 2776619915 4267603449 5420717051 0552366871 7613524950 8472627720 1586507254 4176805495 3097006802 3361070634 6671321568 0747683053 0434691581 4166278084 3725741855 3208119798 3234664225 8251806896 d = 18 9. 9082144987 1693487065 5508584622 3261603707 3060359795 5904711369 3660963521 2134164579 6641458540 7280873307 0219704667 3532523237 2493962550 5615603769 5502230943 4376267333 6939949461 8148090744 3318637498 4322803613 d = 19 10. 6597876907 1106398605 5147427797 8180563813 8374447621 6784535907 4095819042 1008363429 6294084016 0645975103 6738400470 1840322053 8129793615 8559044778 6635212354 2312110507 4169134196 9151999556 6976852579 0612608599 d = 20 11. 2538923172 1643996694 3345869343 0985661735 4842509098 5608933269 5949460201 2336553716 5907540128 3718716686 1740586507 5123951205 6743084610 4493551772 1117397383 0238697136 5154296165 5733074881 0608193795 2939245295 (Python) d = 2 while d < 20: d = d+1 a, n, nmax = 0, 0, 150000 while n < nmax: n, a = n+1, (a*d)//(d-1)+1 num, den, pos = a*(d-1)**n, d**n, 0 dig, num = num//den, (num%den)*10 s = str(dig)+"." print() print("d =", d) while pos <= 200: if pos%50 == 0: print(s) s = "" elif pos%10 == 0: s = s+" " dig, num, pos = num//den, (num%den)*10, pos+1 s = s+str(dig)