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 A079611 Waring's problem: conjectured values for G(n), the smallest number m such that every sufficiently large number is the sum of at most m n-th powers of positive integers. 8
 1, 4, 4, 16, 6, 9, 8, 32, 13, 12, 12, 16, 14, 15, 16, 64, 18, 27, 20, 25 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The only certain values are G(1) = 1, G(2) = 4 and G(4) = 16. See A002804 for the simpler problem of Waring's original conjecture, which does not restrict the bound to "sufficiently large" numbers. - M. F. Hasler, Jun 29 2014 REFERENCES G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 395 (shows G(4) >= 16). R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003. LINKS H. Davenport, On Waring's problem for fourth powers, Annals of Mathematics, 40 (1939), 731-747. [Shows that G(4) <= 16.) Wikipedia, Waring's Problem. Trevor D. Wooley, On Waring's problem for intermediate powers, arXiv:1602.03221 [math.NT], 2016. EXAMPLE It is known that every sufficiently large number is the sum of 16 fourth powers, and 16 is the smallest number with this property, so a(4) = G(4) = 16. (The numbers 16^k*31 are not the sum of fewer than 16 fourth powers.) CROSSREFS Cf. A002376, A002377, A002804, A174406. Sequence in context: A127473 A289625 A040004 * A246763 A319070 A227074 Adjacent sequences:  A079608 A079609 A079610 * A079612 A079613 A079614 KEYWORD nonn,hard,more AUTHOR N. J. A. Sloane, Jan 28 2003; entry revised Jun 29 2014. STATUS approved

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Last modified April 11 06:11 EDT 2021. Contains 342886 sequences. (Running on oeis4.)