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A002804 (Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n) - 2.
(Formerly M3361 N1353)
12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

g(n) is the smallest number s such that every natural number is the sum of at most s n-th 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).

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 one-line 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 counter-example to this inequality is known. - M. F. Hasler, Jun 29 2014

REFERENCES

L. E. Dickson, The Waring Problem and its generalizations, Bulletin of the AMS, 42 (1936) 833-842.

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.

Kubina, Jeffrey M., and Marvin C. Wunderlich. Extending Waring’s conjecture to 471,600,000,  Math. Comp., 55, no. 192 (1990): 815-820.

K. Mahler, On the fractional parts of the powers of a rational number (II), Mathematika, 4 (1957) 122-124 Math. Rev. 20:33.

S. Pillai, On Waring's Problem, Journal of Indian Math. Soc., 2 (1936), 16-44

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 138.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, 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. 285-324 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

A. V. Kumchev and D. I. Tolev, An invitation to additive number theory

M. Waldschmidt, Open Diophantine problems

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 *)

PROG

(PARI) a(n)=2^n+(3^n>>n)-2 \\ Charles R Greathouse IV, Feb 01 2013

CROSSREFS

Cf. A174406, A002376, A002377, A079611, A174420.

Sequence in context: A023611 A192979 A232623 * A133649 A177144 A101353

Adjacent sequences:  A002801 A002802 A002803 * A002805 A002806 A002807

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 27 04:09 EST 2014. Contains 250155 sequences.