This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337844 #19 Oct 14 2020 11:02:23
%S A337844 1,11,111,1112,11124,1112410,111241020,11124102080,11124102080130,
%T A337844 11124102080130102,11124102080130102180,11124102080130102180480,
%U A337844 11124102080130102180480160,11124102080130102180480160116
%N A337844 a(1) = 1; a(n) = a(n-1) concatenated with the smallest number k>0 such that tau(a(n)) > tau(a(n-1)).
%C A337844 This sequence is infinite because k can be of the form m*10^n for some m and n > 0.
%F A337844 a(1) = 1; a(n) = a(n-1) concatenated with the smallest number k>0, such that A000005(a(n)) > A000005(a(n-1)).
%e A337844 a(4) = 1112, because a(3) = 111 and tau(111) = 4, so a(n) must be equal to 111//k ("//" denotes concatenation) and tau(a(4)) > 4, therefore the smallest k that satisfies this condition is 2.
%t A337844 f[1] = {1, 1}; f[n_] := f[n] = Module[{k = 1, t = f[n - 1][[1]], d = IntegerDigits[f[n - 1][[2]]]}, While[(t1 = DivisorSigma[0, (n1 = FromDigits @ Join[d, IntegerDigits[k]])]) <= t, k++]; {t1, n1}]; a[n_] := f[n][[2]]; Array[a, 14] (* _Amiram Eldar_, Sep 26 2020 *)
%o A337844 (PARI) a(n) = if(n == 1, l = m = vector(100); return(l[1] = 1), for(k = 1,oo, j = eval(concat(Str(l[n-1]),k)); if((m[n] = numdiv(j)) > m[n-1], return(l[n] = j))))
%o A337844 for(k=1,10,print1(a(k),", "));
%Y A337844 Cf. A000005.
%K A337844 base,nonn,less
%O A337844 1,2
%A A337844 _Eder Vanzei_, Sep 25 2020