0,2

These numbers agree with the asymptotic formula c*sqrt(x), with c=2.1732...(A090699). - T. D. Noe (noe(AT)sspectra.com), May 09 2006

Filaseta & Trifonov write that a result of Bateman & Grosswald (1958) implies that the asymptotic expansion of the number of powerful numbers up to x is zeta(3/2)/zeta(3) * x^1/2 + zeta(2/3)/zeta(2) * x^1/3 + o(x^1/6). This approximates the series very closely: up to a(24), all absolute errors are less than 75, and up to a(27) all are below 300. - Charles R Greathouse IV, Sep 23 2008

Michael Filaseta and Ognian Trifonov, "The distribution of squarefull numbers in short intervals", Acta Arithmetica 67 (1994), pp. 323-333.

Eric Weisstein's World of Mathematics, Powerful Number

Charles R Greathouse IV, Home Page [in lieu of email address]

nMax=10^12; lst={}; Do[lst=Join[lst, i^3 Range[Sqrt[nMax/i^3]]^2], {i, nMax^(1/3)}]; lst=Union[lst]; k=1; Table[While[lst[[k]]<10^n, k++ ]; If[lst[[k]]==10^n, k, k-1], {n, 0, 12}] - T. D. Noe (noe(AT)sspectra.com), May 09 2006

(PARI) sum(k=1, n^(1/3)+.01, if(issquarefree(k), sqrtint(n\k^3))) - Charles R Greathouse IV, Sep 23 2008

Cf. A001694, A090699.

Adjacent sequences: A118893 A118894 A118895 this_sequence A118897 A118898 A118899

Sequence in context: A017948 A112872 A000651 this_sequence A060898 A045501 A088655

nonn,new

Eric Weisstein (eric(AT)weisstein.com), May 05, 2006

More terms from T. D. Noe (noe(AT)sspectra.com), May 09 2006

Terms from a(13) on from Charles R Greathouse IV, Sep 23 2008