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A322522 a(n) is the minimal absolute difference between n and each of the powers of the previous terms; a(1) = 1.

%I A322522 #48 Mar 23 2024 21:05:30
%S A322522 1,1,2,0,1,2,1,0,1,2,3,3,3,2,1,0,1,2,3,4,5,3,2,1,0,1,0,1,2,2,1,0,1,2,
%T A322522 3,4,5,6,3,4,5,6,7,5,4,3,2,1,0,1,2,3,4,5,6,7,7,6,5,4,3,2,1,0,1,2,3,4,
%U A322522 5,6,7,8,8,7,6,5,4,3,2,1,0,1
%N A322522 a(n) is the minimal absolute difference between n and each of the powers of the previous terms; a(1) = 1.
%C A322522 a(n) <= ceiling(sqrt(n)); if n is between k^2 and (k+1)^2, we have min(n - k^2, (k+1)^2 - n) <= k < ceiling(sqrt(n)).
%C A322522 Using the fact that the density of the nontrivial powers over the integers is 0, and that the density of cubes and higher powers among the nontrivial powers is 0, we can show that there are an infinite number of integers i such that i is not a nontrivial power, and there is no cube or higher power between (i-1)^2 and i^2. We then have a((i-1)^2 + i) = i = ceiling(sqrt((i-1)^2 + i)). Therefore there are infinitely many numbers n such that a(n) = ceiling(sqrt(n)).
%C A322522 a(n) - a(n-1) <= 1.
%C A322522 For the first 10000 terms, indices for a(n) = 1 correspond to 85 of 90 values of A227802(m). - _Bill McEachen_, Feb 26 2024
%H A322522 Gabin Kolly, <a href="/A322522/b322522.txt">Table of n, a(n) for n = 1..10000</a>
%F A322522 Let b(n) be the first time that n appears in the sequence; then b(n) ~ n^2.
%e A322522 For n = 4, we have a(4) = 0, because a(3) = 2, and 2^2 - 4 = 0.
%e A322522 For n = 6, we have a(6) = 0, because there are only 0, 1 and 2 in the first 5 terms, and therefore the closest power is 2^2 = 4 or 2^3 = 8, with an absolute difference of 2.
%t A322522 comparePowers[n_, m_] :=
%t A322522 If[n <= 1, m - n, a = n; While[a < m, a *= n];
%t A322522   Min[m - a/n, a - m]]; list = {1}; cleanList = {1}; Do[
%t A322522 list = Append[list,
%t A322522    Min[comparePowers[#, Length[list] + 1] & /@ cleanList]];
%t A322522 If[Last[list] > Last[cleanList],
%t A322522   cleanList = Append[cleanList, Last[list]]], 9999]; Print[list]
%Y A322522 Cf. A301573 (distance from n to the nearest nontrivial power).
%K A322522 nonn,easy,look,nice,hear
%O A322522 1,3
%A A322522 _Gabin Kolly_, Aug 28 2019