# Lexicographically smallest sequence of integers not in the OEIS

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.

• It probably shouldn't - lest all matter of self-referential and changing issues arise. – Parcly Taxel Oct 20 '18 at 20:24
• @ParclyTaxel: My final remark is tongue-in-cheek. :-) – Joseph O'Rourke Oct 20 '18 at 20:25
• 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 $a_{105} = 2$. – Travis Willse Oct 20 '18 at 20:51

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.