A301573 Distance to nearest perfect power n^k, k>=2 (A001597).

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, 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, 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

Offset

0,7

Comments

Differs from A061670 at n=36.

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, ...

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).

b(n) is A366928(n). - Dmitry Kamenetsky, Oct 28 2023

Links

Altug Alkan, Table of n, a(n) for n = 0..10000

Index entries for sequences related to distance to nearest element of some set

Formula

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

Example

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

Prog

(PARI) isA001597(n) = {ispower(n) || n==1}
a(n) = {my(k=0); while(!isA001597(n+k) && !isA001597(n-k), k++); k; }
(Python)
from itertools import count
from sympy import perfect_power
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

Crossrefs

Cf. A001597, A061670, A366928.

Sequence in context: A058755 A128519 A303979 * A061670 A236412 A236263

Adjacent sequences: A301570 A301571 A301572 * A301574 A301575 A301576

Keyword

nonn

Author

Rémy Sigrist and Altug Alkan, Mar 23 2018

Status

approved