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A075772
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Difference between the n-th perfect power and the closest perfect power.
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6
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3, 3, 1, 1, 7, 2, 2, 4, 4, 13, 15, 17, 19, 4, 3, 3, 16, 25, 20, 9, 9, 13, 13, 33, 19, 18, 18, 39, 41, 28, 17, 17, 47, 49, 51, 53, 55, 57, 59, 39, 24, 24, 65, 67, 69, 35, 35, 38, 75, 77, 79, 47, 36, 36, 85, 87, 23, 23, 68, 10, 10, 12, 95, 97, 99, 101, 40, 40, 65, 107, 100, 11, 11
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OFFSET
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1,1
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COMMENTS
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Let {b(n)} be the sequence of perfect powers (A001597); then a(n) = min { b(n)-b(n-1), b(n+1)-b(n) }.
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LINKS
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FORMULA
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EXAMPLE
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The perfect powers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, etc. The 7th is 27. This is 2 larger than the 6th (25) and 5 smaller than the 8th (32). So a(7)=2.
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MATHEMATICA
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pp = {-2, 1}; Do[ If[ !PrimeQ[n] && Apply[GCD, Last[ Transpose[ FactorInteger[n]]]] > 1, pp = Append[pp, n]], {n, 2, 10^4}]; Table[ Min[pp[[n + 1]] - pp[[n]], pp[[n + 2]] - pp[[n + 1]]], {n, 1, 75}]
perfPQ[n_]:=GCD@@FactorInteger[n][[All, 2]]>1; Join[{3, 3}, Min[ Differences[ #]]&/@Partition[Select[Range[5000], perfPQ], 3, 1]] (* Harvey P. Dale, May 04 2021 *)
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PROG
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(PARI) for(n=L=3+P=-2, 99, ispower(n)&&print1(min(-P+P=L, -L+L=n)", ")) \\ Note: ispower(1)=0. - M. F. Hasler, May 08 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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