E. Angelini, <a href="http://list.seqfan.eu/pipermailoldermail/seqfan/2012-September/010124.html">Strings resurrection</a>, SeqFan mailing list, Sep 08 2012

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In A216557 is it is explained why a(10)=111 and all numbers having this or 222,...,999 as substring are in this list.

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0, 20, 30, 40, 50, 60, 70, 80, 90, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159

1,21

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.

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.

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.

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.

In A216557 is is explained why a(10)=111 and all numbers having this or 222,...,999 as substring are in this list.

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.

0, 20, 30, 40, 50, 60, 70, 80, 90, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164

Cf. A216556, A216557, A216587, A216589.

allocated for M. F. Hasler

Indices n for which A216557(n)=0, i.e., n does not reappear as substring in its orbit under A216556.

0, 20, 30, 40, 50, 60, 70, 80, 90, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164

1,2

E. Angelini asked on the SeqFan list whether n=127 is in this sequence.

E. Angelini, <a href="http://list.seqfan.eu/pipermail/seqfan/2012-September/010124.html">Strings resurrection</a>, SeqFan mailing list, Sep 08 2012

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nonn,base

M. F. Hasler, Sep 09 2012

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allocated for M. F. Hasler

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