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The factorisation of (10^k-1)/9 is so chaotic
11=11
111=3*37
1111=11*101
11111=41*271
111111=3*7*11*13*37
1111111=239*4649
11111111=11*73*101*137
111111111=3*3*37*3336667
1111111111=11*41*271*9091
11111111111=21649*513239
111111111111=3*7*11*13*37*101*9901
1111111111111=53*79*265371653
11111111111111=11*239*4649*909091
111111111111111=3*31*37*41*271*2906161
1111111111111111=11*17*73*101*137*5882353
11111111111111111=2071723*5363222357
111111111111111111=3*3*7*11*13*19*37*52579*333667
1111111111111111111=1111111111111111111
11111111111111111111=11*41*101*271*3541*9091*27961
111111111111111111111=3*37*43*239*1933*4649*10838689
1111111111111111111111=11*11*23*4093*8779*21649*513239
11111111111111111111111=11111111111111111111111
111111111111111111111111=3*7*11*13*37*73*101*137*9901*99990001
1111111111111111111111111=41*271*21401*25601*182521213001
11111111111111111111111111=11*53*79*859*265371653*1058313049
111111111111111111111111111=3*3*3*37*757*333667*440334654777631
1111111111111111111111111111=11*29*101*239*281*4649*909091*121499449
11111111111111111111111111111=3191*16763*43037*62003*77843839397
111111111111111111111111111111=3*7*11*13*31*37*41*211*241*271*2161*9091*2906161
.........
new numbers:
31->2791;6943319;573364150637906043579;
32->353;449;641;1409;69857;
33->67;1344628210313298373;
34->103;4013;21993833369;
35->71;123551;102598800232111471;
36->999999000001;
37->2028119;247629013;2112394296770203368013;
38->909090909090909091;
39->900900900900990990990991;
40->1676321;5964848081;
41->83;1231;538987;201763709900322803748657942361
42->7;127;2689;459691;
43->173;1527791;1963506722254397;2140992015395526641;
44->89;1052788969;1056689261
45->238681;4185502830133110721
46->47;139;2531;54979718449197;
47->35121409;3163629087634585250014061540387626382279;
48->9999999900000001;
49->505885997;1976730144598190963568023014679333
50->251;5051;78875943472201;
51->613;210631;52986961;13168164561429877;
52->521;1910381976777332243781;
53->107;1659431;1325815267337711173;47198858799491425660200071;
54->70541929;14175966169;440334654777631;
55->1321;62921;83251631;1300635692678058358830121
56->7841;127522001020150503761;
57->21319;10749631;3931123022305129377976519;
58->59;154083204930662557781201849;
59->2559647034361;4340876285657460212144534289928559826755746751;
60->61;4188901;39526741;

Those datas taken from WOLFRAM prime numbers calculator

That's is misterious the order of the prime numbers, the distribution of prime numbers is f(x)=1/ln(x)