%I #43 Feb 16 2025 08:32:33

%S 1,7,23,79,223,703,2175,6399,19455,58367,176127,528383,1589247,

%T 4767743,14319615,42991615,129105919,387186687,1161822207,3486515199,

%U 10458497023,31377588223,94136958975,282427654143,847282962431,2541815332863

%N Waring's problem: least positive integer requiring maximum number of terms when expressed as a sum of positive n-th powers.

%C a(n) = (Q-1)*(2^n) + (2^n-1)*(1^n) is a sum of Q + 2^n - 2 terms, Q = trunc(3^n / 2^n).

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 393.

%H T. D. Noe, <a href="/A018886/b018886.txt">Table of n, a(n) for n = 1..200</a>

%H P. Pollack, <a href="https://web.archive.org/web/20130509125813/http://alpha01.dm.unito.it/personalpages/cerruti/ac/notes.pdf">Analytic and Combinatorial Number Theory</a> Course Notes, exercise 7.1.1. p. 277.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>.

%F a(n) = 2^n*floor((3/2)^n) - 1 = 2^n*A002379(n) - 1.

%e a(3) = 23 = 16 + 7 = 2*(2^3) + 7*(1^3) is a sum of 9 cubes;

%e a(4) = 79 = 64 + 15 = 4*(2^4) + 15*(1^4) is a sum of 19 biquadrates.

%p A018886 := proc(n)

%p 2^n*floor((3/2)^n)-1

%p end proc: # _R. J. Mathar_, May 07 2015

%t a[n_]:=-1+2^n*Floor[(3/2)^n]

%t a[Range[1,20]] (* _Julien Kluge_, Jul 21 2016 *)

%o (Python)

%o def a(n): return (3**n//2**n-1)*2**n + (2**n-1)

%o print([a(n) for n in range(1, 27)]) # _Michael S. Branicky_, Dec 17 2021

%o (Python)

%o def A018886(n): return (3**n&-(1<<n))-1 # _Chai Wah Wu_, Jun 25 2024

%Y Cf. A079611, A185187.

%Y Cf. A000079, A002379.

%K nonn,easy,nice,changed

%O 1,2

%A _N. J. A. Sloane_