Given that the sequence of noncototients, i.e numbers not expressible as $n-\phi{(n)}$, probably has positive lower density, by Szemerédi theorem it should contain arithmetic progressions of any length. So this led me to the question of whether this sequence contains arbitrarily long subsequences of consecutive even numbers, i.e for every $k$, there is $x$, such that every even in $[2x+2;2x+2k]$ is noncototient. Plenty of examples with two consecutive terms such as [50,52] appear on OEIS, but I am not aware of any three terms. EDIT: Proving that It contains arithmetic progressions of any length (unconditionally) would be nice.
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4$\begingroup$ oeis.org/A005278/b005278.txt $532,534,536$. $1462,1464,1466$. $2072,2074,2076$. Four in a row, from $2314$. $\endgroup$Gerry Myerson– Gerry Myerson2025-12-05 20:55:48 +00:00Commented Dec 5 at 20:55
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1$\begingroup$ There are also longer ones: first run of five $4628,4630,4632,4634,4636$; first run of six $22578,22580,22582,22584,22586,22588$ $\endgroup$მამუკა ჯიბლაძე– მამუკა ჯიბლაძე2025-12-06 06:55:20 +00:00Commented Dec 6 at 6:55
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