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 A216603 #11 Dec 23 2024 14:53:43 %S A216603 20,30,40,50,60,70,80,90,100,111,112,113,114,115,116,117,118,119,120, %T A216603 122,123,124,125,126,127,128,129,130,131,133,134,135,136,137,138,139, %U A216603 140,141,142,144,145,146,147,148,149,150,151,152,153,155,156,157,158,159 %N A216603 Indices n for which A216557(n)=0, i.e., n does not reappear as substring in its orbit under A216556. %C A216603 E. Angelini asked on the SeqFan list whether n=127 is in this sequence. The proof of the affirmative answer was given in the second reply to the post. (But "Lemma 2" should read: There is no n in the range of A216556 such that ...". Indeed, 2016,...,9016 also yield 127 as substring, but are not in A216556.) The same reasoning can be applied to terms 122,...,129. %H A216603 E. Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2012-September/010124.html">Strings resurrection</a>, SeqFan mailing list, Sep 08 2012 %e A216603 The number 0 is not in this sequence, because repeated application of A216556 yields 0 > 1 > ... > 9 > 10 which contains 0 as substring, after A216557(0)=10 iterations. %e A216603 a(1)=20 is the least number which does not occur as a substring in its orbit under A216556. This is the case since no term in (the range of) A216556 may have "20" as a substring. %e A216603 As explained in A216556 (and obvious from the definition), the digit "0" can only occur preceded by the digit "1". This also explains a(2)=30 through a(9)=100. %e A216603 In A216557 it is explained why a(10)=111 and all numbers having this or 222,...,999 as substring are in this list. %e A216603 To see why a(11)=112 and the following terms are in this list, observe that 112 itself has no preimage under A216556 (cf. A216587), and consider all possibilities for "patterns" in which it might occur: Obviously not followed by a digit 0, but possibly by a digit among 1,...,9; and maybe preceded by a digit among 2,...,9. However, all these cases lead to a preimage which has "00" as substring, and is therefore not in the range of A216556. %Y A216603 Cf. A216556, A216557, A216587, A216589. %K A216603 nonn,base %O A216603 1,1 %A A216603 _M. F. Hasler_, Sep 09 2012