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A sequence $a_i$ ($i=1,\ldots$) is lexicographically smaller than sequence $b_i$ if either $a_1 < b_1$, or $a_j = b_j$ for $j=1,\ldots, k$ and $a_{k+1} < b_{k+1}$.

If I asked for the lexicographically smallest sequence of natural numbers not in the OEIS, then I think it would start $1,1,1,1,1,1,1,1,1,1,1,2$—eleven $1$'s followed by a $2$—because A055642 starts with ten $1$'s followed by a $2$.

But what about integer sequences? After seeing @RossMillikan's answer, what I should ask for is the largest of all those sequences smaller than any sequence in the OEIS.

Of course once identified, it could be added to the OEIS.

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    $\begingroup$ It probably shouldn't - lest all matter of self-referential and changing issues arise. $\endgroup$ Oct 20, 2018 at 20:24
  • $\begingroup$ @ParclyTaxel: My final remark is tongue-in-cheek. :-) $\endgroup$ Oct 20, 2018 at 20:25
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    $\begingroup$ For the reason Ross Millikan explains in his answer, there is no lexicographically smallest sequence after $(1, 1, 1, \ldots)$. It is, however, meaningful to ask what sequence is lexicographically second smallest. One candidate is oeis.org/A160338, Height (maximum absolute value of coefficients) of the n-th cyclotomic polynomial: $a_1 = a_2 = \cdots = a_{104} = 1$, but $a_{105} = 2$. $\endgroup$ Oct 20, 2018 at 20:51

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Given that you ask for naturals which do not include $0$, the first sequence lexicographically is OEIS A000012, which is all $1$'s. There is no sequence which is the next one lexicographically after this. You suggest starting with eleven $1$'s and a $2$, but then I suggest starting with twelve $1$'s and a $2$, then someone else will suggest a hundred $1$'s and a $2$, and so on.

The same problem occurs for integer sequences. Given any sequence that is missing, there is a lexicographically earlier one missing.

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  • $\begingroup$ I see I asked incorrectly: I meant the maximum of all those smaller than the smallest. May not be repairable now... $\endgroup$ Oct 20, 2018 at 20:57

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