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A119241
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Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.
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9
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0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 2, 0, 0, 3, 1, 0, 1, 0, 1, 1, 0, 2, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 0, 2, 0, 2, 2, 1, 0, 1, 1, 1, 1, 0, 0, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 3, 1, 0, 2, 0, 2, 0, 1, 1, 1, 2, 2, 0
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OFFSET
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1,5
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COMMENTS
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Is there an upper bound on the number of powerful numbers between consecutive squares? Pettigrew conjectures that there is no bound. See A119242.
This sequence is unbounded. For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1980). - Amiram Eldar, Jul 10 2020
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REFERENCES
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József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.
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LINKS
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FORMULA
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Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3/2)/zeta(3) - 1 = A090699 - 1 = 1.173254... - Amiram Eldar, Oct 24 2020
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EXAMPLE
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a(5) = 2 because the two powerful numbers 27 and 32 are between 25 and 36.
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MATHEMATICA
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Powerful[n_] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Table[Length[Select[Range[k^2+1, k^2+2k], Powerful[ # ]&]], {k, 130}]
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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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