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Search: id:A118896
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| A118896 |
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Number of powerful numbers <= 10^n. |
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+0
1
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| 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, 6840384, 21663503, 68575557, 217004842, 686552743, 2171766332, 6869227848, 21725636644, 68709456167, 217293374285, 687174291753, 2173105517385, 68722847672628
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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
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REFERENCES
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Michael Filaseta and Ognian Trifonov, "The distribution of squarefull numbers in short intervals", Acta Arithmetica 67 (1994), pp. 323-333.
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LINKS
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Eric Weisstein's World of Mathematics, Powerful Number
Charles R Greathouse IV, Home Page [in lieu of email address]
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MATHEMATICA
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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
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PROGRAM
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(PARI) sum(k=1, n^(1/3)+.01, if(issquarefree(k), sqrtint(n\k^3))) - Charles R Greathouse IV, Sep 23 2008
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CROSSREFS
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Cf. A001694, A090699.
Sequence in context: A112872 A162482 A000651 this_sequence A145211 A060898 A045501
Adjacent sequences: A118893 A118894 A118895 this_sequence A118897 A118898 A118899
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KEYWORD
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nonn
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com), May 05, 2006
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EXTENSIONS
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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
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