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%I M3361 N1353 #84 Apr 18 2024 11:38:55
%S 1,4,9,19,37,73,143,279,548,1079,2132,4223,8384,16673,33203,66190,
%T 132055,263619,526502,1051899,2102137,4201783,8399828,16794048,
%U 33579681,67146738,134274541,268520676,536998744,1073933573,2147771272,4295398733,8590581749
%N (Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n) - 2.
%C 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.
%C 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).
%C Mahler showed that there are only finitely many n's for which this formula fails. - _Tomohiro Yamada_, Sep 23 2017
%C 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
%D Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers (Basic Books 1996) 252-257.
%D G. H. Hardy, Collected Papers. Vols. 1-, Oxford Univ. Press, 1966-; see vol. 1, p. 668.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 337.
%D S. Pillai, On Waring's Problem, Journal of Indian Math. Soc., 2 (1936), 16-44
%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 138.
%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 239.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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.
%D Edward Waring, Meditationes algebraicae, Cantabrigiae: typis Academicis excudebat J. Archdeacon, 1770.
%H T. D. Noe, <a href="/A002804/b002804.txt">Table of n, a(n) for n = 1..200</a>
%H Brennan Benfield and Oliver Lippard, <a href="https://arxiv.org/abs/2404.08193">Integers that are not the sum of positive powers</a>, arXiv:2404.08193 [math.NT], 2024.
%H Leonard E. Dickson, <a href="http://dx.doi.org/10.1090/S0002-9904-1936-06432-3">The Waring Problem and its generalizations</a>, Bulletin of the AMS, 42 (1936) 833-842.
%H Jeffrey M. Kubina and Marvin C. Wunderlich, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-1035936-6">Extending Waring’s conjecture to 471,600,000</a>, Math. Comp., 55, no. 192 (1990): 815-820.
%H A. V. Kumchev and D. I. Tolev, <a href="http://arxiv.org/abs/math/0412220">An invitation to additive number theory</a>, arXiv:math/0412220 [math.NT], 2004.
%H Feihu Liu and Guoce Xin, <a href="https://arxiv.org/abs/2210.02722">On Frobenius Formulas of Power Sequences</a>, arXiv:2210.02722 [math.CO], 2022. See p. 22.
%H Kurt Mahler, <a href="http://dx.doi.org/10.1112/S0025579300001170">On the fractional parts of the powers of a rational number (II)</a>, Mathematika, 4 (1957) 122-124 Math. Rev. 20:33.
%H Ramin Takloo-Bighash, <a href="https://doi.org/10.1007/978-3-030-02604-2_10">What about geometry?</a>, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 165-185.
%H Michel Waldschmidt, <a href="http://arxiv.org/abs/math/0312440">Open Diophantine problems</a>, arXiv:math/0312440 [math.NT], 2003-2004.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Waring%27s_problem">Waring's Problem</a>.
%p A002804 := n->2^n+floor( (3/2)^n ) -2;
%t a[n_] := 2^n + Floor[(3/2)^n] - 2; Array[a, 31] (* _Robert G. Wilson v_, Oct 29 2013 *)
%t 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 *)
%o (PARI) a(n)=2^n+(3^n>>n)-2 \\ _Charles R Greathouse IV_, Feb 01 2013
%o (Magma) [2^n+Floor((3/2)^n)-2: n in [1..40]]; // _Vincenzo Librandi_, Aug 15 2015
%Y Cf. A002376, A002377, A079611, A174406, A174420, A297446 (for info on Mathematica functions).
%K nonn,easy,changed
%O 1,2
%A _N. J. A. Sloane_
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