

A002804


(Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n)  2.
(Formerly M3361 N1353)


23



1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, 2102137, 4201783, 8399828, 16794048, 33579681, 67146738, 134274541, 268520676, 536998744, 1073933573, 2147771272, 4295398733, 8590581749
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OFFSET

1,2


COMMENTS

g(n) is the smallest number s such that every natural number is the sum of at most s nth powers of natural numbers.
It is known (Kubina and Wunderlich, 1990) that g(n) = 2^n + floor((3/2)^n)  2 for all n <= 471600000. This formula is conjectured to be correct for all n (see A174420).
Mahler showed that there are only finitely many n's for which this formula fails.  Tomohiro Yamada, Sep 23 2017
This sequence (which corresponds to Waring's original conjecture) is much easier to compute than A079611, the problem of finding the minimal s = G(n) for almost all (= sufficienly large) integers. See Wikipedia for a oneline proof that this value for g(n), conjectured by J. A. Euler in 1772, is indeed a lower bound; it is known to be tight if 2^n*frac((3/2)^n) + floor((3/2)^n) <= 2^n, and no counterexample to this inequality is known.  M. F. Hasler, Jun 29 2014


REFERENCES

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers (Basic Books 1996) 252257.
G. H. Hardy, Collected Papers. Vols. 1, Oxford Univ. Press, 1966; see vol. 1, p. 668.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 337.
S. Pillai, On Waring's Problem, Journal of Indian Math. Soc., 2 (1936), 1644
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 138.
P. Ribenboim, The Book of Prime Number Records. SpringerVerlag, NY, 2nd ed., 1989, p. 239.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
Edward Waring, Meditationes algebraicae, Cantabrigiae: typis Academicis excudebat J. Archdeacon, 1770.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..200
L. E. Dickson, The Waring Problem and its generalizations, Bulletin of the AMS, 42 (1936) 833842.
Jeffrey M. Kubina and Marvin C. Wunderlich, Extending Waringâ€™s conjecture to 471,600,000, Math. Comp., 55, no. 192 (1990): 815820.
A. V. Kumchev and D. I. Tolev, An invitation to additive number theory, arXiv:math/0412220 [math.NT], 2004.
K. Mahler, On the fractional parts of the powers of a rational number (II), Mathematika, 4 (1957) 122124 Math. Rev. 20:33.
M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 20032004.
Eric Weisstein's World of Mathematics, Waring's Problem.
Wikipedia, Waring's Problem.


MAPLE

A002804 := n>2^n+floor( (3/2)^n ) 2;


MATHEMATICA

a[n_] := 2^n + Floor[(3/2)^n]  2; Array[a, 31] (* Robert G. Wilson v, Oct 29 2013 *)
x[n_] := (1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; a[n_] := 2^n + x[n]  2; Array[a, 31] (* Fred Daniel Kline, Jan 11 2018 *)


PROG

(PARI) a(n)=2^n+(3^n>>n)2 \\ Charles R Greathouse IV, Feb 01 2013
(MAGMA) [2^n+Floor((3/2)^n)2: n in [1..40]]; // Vincenzo Librandi, Aug 15 2015


CROSSREFS

Cf. A002376, A002377, A079611, A174406, A174420, A297446 (for info on Mathematica functions).
Sequence in context: A023611 A192979 A232623 * A133649 A177144 A266370
Adjacent sequences: A002801 A002802 A002803 * A002805 A002806 A002807


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



