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Distance to nearest perfect power n^k, k>=2 (A001597).
3

%I #37 Nov 12 2023 21:52:53

%S 1,0,1,1,0,1,2,1,0,0,1,2,3,3,2,1,0,1,2,3,4,4,3,2,1,0,1,0,1,2,2,1,0,1,

%T 2,1,0,1,2,3,4,5,6,6,5,4,3,2,1,0,1,2,3,4,5,6,7,7,6,5,4,3,2,1,0,1,2,3,

%U 4,5,6,7,8,8,7,6,5,4,3,2,1,0,1,2,3,4,5,6,7,8,9,9,8,7,6,5,4,3,2,1,0

%N Distance to nearest perfect power n^k, k>=2 (A001597).

%C Differs from A061670 at n=36.

%C Let b(n) be the smallest t such that a(t) = n. Initial values of b(n) are 1, 0, 6, 12, 20, 41, 42, 56, 72, 90, 110, 155, 156, 182, 270, 271, 272, 306, 379, ...

%C The b(n) sequence determines the positions of certain humps of a(n) sequence. See scatterplot of this sequence in order to observe general structure of a(n).

%C b(n) is A366928(n). - _Dmitry Kamenetsky_, Oct 28 2023

%H Altug Alkan, <a href="/A301573/b301573.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Di#distance_to_the_nearest">Index entries for sequences related to distance to nearest element of some set</a>

%F a(n) = 0 iff n belongs to A001597.

%e a(20) = a(21) = 4 because 16 is the nearest perfect power to 20 and 25 is the nearest perfect power to 21 (20 - 16 = 25 - 21 = 4).

%e a(36) = 0 because 36 is a square.

%o (PARI) isA001597(n) = {ispower(n) || n==1}

%o a(n) = {my(k=0); while(!isA001597(n+k) && !isA001597(n-k), k++); k;}

%o (Python)

%o from itertools import count

%o from sympy import perfect_power

%o def A301573(n): return next(k for k in count(0) if perfect_power(n+k) or perfect_power(n-k) or n-k==1 or n+k==1) # _Chai Wah Wu_, Nov 12 2023

%Y Cf. A001597, A061670, A366928.

%K nonn

%O 0,7

%A _Rémy Sigrist_ and _Altug Alkan_, Mar 23 2018