Tuesday, June 14, 2022

Exceptional fractions

I am assuming that you are acquainted with divisors and know how to calculate and count them. I want to introduce here the concept of exceptional fractions:

4/5, 4/7, 6/7, 4/11, 6/11, 36/143, 8/11, 9/11, 6/13, 8/11, 9/11, ...

The use of "exceptional" comes from OEIS A072066 wherein are defined "exceptional (or extraordinary) numbers: m such that A005179(m) < A037019(m)". The fractions are in fact the reduced ratios A005179(m)/A037019(m) which are therefore numbers between zero and one. There is one such fraction for each term in A072066, so a better approach to presenting these fractions is to couple them with those terms:

8 4/5
16 4/7
24 6/7
32 4/11
48 6/11
64 36/143
72 8/11
80 9/11
96 6/13
108 8/11
112 9/11
...

You will notice that the fraction at 108 repeats the fraction at 72 and that the fraction at 112 repeats the fraction at 80. As we chart the fractions [15.7 MB] for larger exceptional numbers [to 80000000], an overwhelming majority of them will be such repeats.

Are these fractions good for anything? Absolutely. Let's do a little refresh. A037019 terms are "an easy way to produce a number with exactly n divisors". A005179 terms, on the other hand, are the "smallest number with exactly n divisors". Mostly, the easy way also produces the smallest number. But not always. A072066 terms are of course the exceptions. The easy way of finding a number with exactly 8 divisors produces 30. But the smallest number with exactly 8 divisors is actually 24. What the fraction for the first exception, 8, allows one to do is determine the 24 as 4/5 of that 30. This becomes increasingly useful as our exceptions increase in size. What is the smallest number with exactly 3603600 divisors? Looking at our chart, 3603600 isn't even in there as an exceptional number, so the easy calculation, which produces 2549066103582535692163008000000, also produces the smallest number. What is the smallest number with exactly 10000000 divisors? This time, 10000000 is in the chart alongside 32/43. Our easy calculation produces 1740652905587144828469399739530000 and multiplying this by 32/43 yields 1295369604157875221186530038720000. This is the smallest number.

My most common exceptional fractions are 9/11, 9/13, 9/17, 15/17, 9/19, 25/29, 15/19, 9/23, 25/31, 15/23, 9/29, 27/29, 21/23, 25/37, 15/29, 35/37, 225/899, 27/31, 21/29, 25/41, ...

Here are the exceptional fractions for small powers of two:

2^  3 4/5
2^  4 4/7
2^  5 4/11
2^  6 36/143
2^  7 36/221
2^  8 36/323
2^  9 576/7429
2^ 10 576/12673
2^ 11 576/20677
2^ 12 576/33263
2^ 13 14400/1363783
2^ 14 14400/2022161
2^ 15 14400/3065857
2^ 16 14400/4391633
2^ 17 14400/6319667
2^ 18 14400/8965109
2^ 19 14400/12780049
2^ 20 705600/907383479
2^ 21 705600/1249792339
2^ 22 705600/1673450759
2^ 23 705600/2276990377
2^ 24 705600/3024658859
2^ 25 705600/4132280413
2^ 26 705600/5717264681
2^ 27 705600/7454155217
2^ 28 57153600/797594608219
2^ 29 57153600/1047443521637
2^ 30 57153600/1329900201629
2^ 31 57153600/1741209542339
2^ 32 57153600/2258400495509
2^ 33 57153600/3003891921211
2^ 34 57153600/3902252121947
2^ 35 57153600/5334271249267
2^ 36 57153600/7128096979109
2^ 37 6915585600/1119111225720113
2^ 38 6915585600/1436339604664397
2^ 39 6915585600/1831058885335529
2^ 40 6915585600/2312213044985741
2^ 41 6915585600/2977598093902501
2^ 42 6915585600/3617082248297669
2^ 43 6915585600/4575249731290429
2^ 44 6915585600/5624351580503521
2^ 45 6915585600/6797529210792599
2^ 46 6915585600/8100049778130103
2^ 47 1168733966400/1709110503185451733
2^ 48 1168733966400/2203073076360437783
2^ 49 1168733966400/2793841275607929479
2^ 50 1168733966400/3534749459194562711
2^ 51 1168733966400/4312024209383942993
2^ 52 1168733966400/5339760549444364639
2^ 53 1168733966400/6532397423431938467
2^ 54 1168733966400/8239355544127721383
2^ 55 1168733966400/10035613150904381021
2^ 56 1168733966400/11835723133129382101
2^ 57 1168733966400/14025592611505743547
2^ 58 1168733966400/16597972042436927953
2^ 59 1168733966400/19732353028991540957
2^ 60 1168733966400/23199963184713903803
2^ 61 1168733966400/27243110295742882889
2^ 62 1168733966400/31801718393038504727
2^ 63 299195895398400/9763127546662820951189
2^ 64 299195895398400/11814523996156176326147
2^ 65 299195895398400/14060631219760012129597
2^ 66 299195895398400/16569591437412356301421
2^ 67 299195895398400/20238135667097748840481
2^ 68 299195895398400/24621847363942026567661
2^ 69 299195895398400/30404914716326986544407
2^ 70 299195895398400/37495813554763668918721
2^ 71 86467613770137600/13236022184831575128308513
2^ 72 86467613770137600/16217515236704899218644219
2^ 73 86467613770137600/19387062188503902323265239
2^ 74 86467613770137600/23252007062096320149768277
2^ 75 86467613770137600/28154986187011199158984591
2^ 76 86467613770137600/34016907601341606554861509
2^ 77 86467613770137600/39977574190096329153598571
2^ 78 86467613770137600/47095243185365705264031551
2^ 79 86467613770137600/54424185928909647869961533
2^ 80 86467613770137600/63780779498349701945026553
2^ 81 86467613770137600/75705797761497238286023019
2^ 82 86467613770137600/88780336650669463282494961
2^ 83 31214808571019673600/38264325096438538674755328191
2^ 84 31214808571019673600/45145647865825305847872090209
2^ 85 31214808571019673600/53133885825998148169479484187
2^ 86 31214808571019673600/62106362588171977939523513179
2^ 87 31214808571019673600/72808764496316496331190750437
2^ 88 31214808571019673600/85536260603641745047182963881
2^ 89 31214808571019673600/99325481456621774475444197353
2^ 90 31214808571019673600/114682538439939854319527838839
2^ 91 31214808571019673600/130945587900860420457749390557
2^ 92 31214808571019673600/149696746072821339855040472737
2^ 93 31214808571019673600/173164644507040362255118076539
2^ 94 31214808571019673600/197271091538182872081816648679
2^ 95 31214808571019673600/227340126276104510782509255637
2^ 96 31214808571019673600/260483105961003573857863680149
2^ 97 31214808571019673600/299290972763320133394249691187
2^ 98 31214808571019673600/347284179977037393092214006923
2^ 99 31214808571019673600/397439006844618285748857605297
2^100 31214808571019673600/466408899572534691084884955457
2^101 31214808571019673600/551027360834074462253632981933
2^102 31214808571019673600/657221070630791168041271029843
2^103 31214808571019673600/772474870073351623397151544471
2^104 31214808571019673600/902542507334162369020491229577
2^105 31214808571019673600/1049596276349911838514664953337
2^106 31214808571019673600/1213661425759316895436796950051
2^107 31214808571019673600/1416340471015345959485884313479
2^108 31214808571019673600/1650078387646562188556246361283
2^109 31214808571019673600/1897115075240481287802670960477
2^110 31214808571019673600/2180050019540208898603069306399
2^111 16512633734069407334400/1323290361860906801452063068984193
2^112 16512633734069407334400/1499402942367349111441986434911849
2^113 16512633734069407334400/1691282660769020844167651975028539
2^114 16512633734069407334400/1879540335755877742441250579071213
2^115 16512633734069407334400/2106554088564758180249430045104681
2^116 16512633734069407334400/2373112778154674856836352651866609
2^117 16512633734069407334400/2672349415680308113740411129860297
2^118 16512633734069407334400/2996551251204782234991414213205567
2^119 16512633734069407334400/3333471834815541395995559593225273
2^120 16512633734069407334400/3704482190798384114605520694663499
2^121 16512633734069407334400/4087917743101388814280883437683761
2^122 16512633734069407334400/4577651649096896292863618225559353
2^123 16512633734069407334400/5105552168762106738498631859478883
2^124 16512633734069407334400/5688567913971482711899780685194253
2^125 16512633734069407334400/6370827274804367186260532339496319
2^126 16512633734069407334400/7214781776474735698818470387701001

This suggests that we can get as close to zero as we like. The numerators are squares and the denominators appear to be the products of consecutive primes. As for approaching one (in the other direction), my largest exceptional fraction is currently 900/901 at 4354560 (thus beating 400/403 at 64800).