A297446 -id:A297446 - cp's OEIS frontend

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

Offset: 1

N. J. A. Sloane

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).
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 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 counterexample to this inequality is known. - M. F. Hasler, Jun 29 2014

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers (Basic Books 1996) 252-257.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30.1, p. 195.
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), 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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 249-250.
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.

T. D. Noe, Table of n, a(n) for n = 1..200
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024.
Leonard E. Dickson, The Waring Problem and its generalizations, Bulletin of the AMS, 42 (1936) 833-842.
Jeffrey M. Kubina and Marvin C. Wunderlich, Extending Waring's conjecture to 471,600,000, Math. Comp., 55, no. 192 (1990): 815-820.
A. V. Kumchev and D. I. Tolev, An invitation to additive number theory, arXiv:math/0412220 [math.NT], 2004.
Feihu Liu and Guoce Xin, On Frobenius Formulas of Power Sequences, arXiv:2210.02722 [math.CO], 2022. See p. 22.
Kurt Mahler, On the fractional parts of the powers of a rational number (II), Mathematika, 4 (1957) 122-124 Math. Rev. 20:33.
Ramin Takloo-Bighash, What about geometry?, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 165-185.
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Eric Weisstein's World of Mathematics, Waring's Problem.
Wikipedia, Waring's Problem.

Cf. A002376, A002377, A079611, A174406, A174420, A297446 (for info on Mathematica functions).

[2^n+Floor((3/2)^n)-2: n in [1..40]]; // Vincenzo Librandi, Aug 15 2015

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

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 *)

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

def A002804(n): return (1<>n)-2 # Chai Wah Wu, Jun 25 2024

[2^n+int((3/2)^n)-2 for n in range(1,34)] # Stefano Spezia, Dec 08 2024

cp's OEIS Frontend

A002804 (Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n) - 2.

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