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 A080440 #13 Dec 23 2021 02:24:22
%S A080440 13,103,10093,100000963,10000000000092653,
%T A080440 100000000000000000000000902060523,
%U A080440 10000000000000000000000000000000000000000000000090002000600051233
%N A080440 a(1) = 13, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).
%C A080440 Conjecture: Only one digit needs to be inserted between each pair of digits of a(n-1) to get a(n); i.e., a(n) contains exactly 2n-1 digits for n > 1.
%C A080440 The conjecture above is false: a(4) = 100000963 has 9 digits instead of 2*4 - 1 = 7. A refined conjecture is: a(n) contains exactly 2^(n-1) + 1 digits for all n > 0. This follows trivially by induction from the initial above conjecture of only one digit needed between each pair, and the fact that we start with 13, a 2-digit number, and holds true at least till a(12). - _Julio Cesar Hernandez-Castro_, Jul 06 2011
%H A080440 Julio Cesar Hernandez-Castro, <a href="/A080440/b080440.txt">Table of n, a(n) for n = 1..12</a>
%t A080440 a[n_] := Block[{d = IntegerDigits[n]}, k = Length[d]; While[k > 1, d = Insert[d, 0, k]; k-- ]; d = FromDigits[d]; e = d; k = 0; While[ !PrimeQ[e], k++; e = d + 10FromDigits[ IntegerDigits[k], 100]]; e]; NestList[a, 13, 6]
%Y A080440 Cf. A080439, A080441, A080442, A080883 - A080914.
%K A080440 base,nonn
%O A080440 1,1
%A A080440 _Amarnath Murthy_, Feb 22 2003
%E A080440 Edited, corrected and extended by _Robert G. Wilson v_, Feb 22 2003