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 A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2. 8
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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 REFERENCES József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178. FORMULA 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 EXAMPLE a(5) = 2 because the two powerful numbers 27 and 32 are between 25 and 36. MATHEMATICA Powerful[n_] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Table[Length[Select[Range[k^2+1, k^2+2k], Powerful[ # ]&]], {k, 130}] CROSSREFS Cf. A001694, A090699, A119242. Sequence in context: A116357 A035168 A255647 * A001878 A056558 A320808 Adjacent sequences:  A119238 A119239 A119240 * A119242 A119243 A119244 KEYWORD nonn AUTHOR T. D. Noe, May 09 2006 STATUS approved

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Last modified May 13 05:02 EDT 2021. Contains 343836 sequences. (Running on oeis4.)