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A301573
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Distance to nearest perfect power n^k, k>=2 (A001597).
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3
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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
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OFFSET
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0,7
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COMMENTS
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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).
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LINKS
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FORMULA
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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