A018886 Waring's problem: least positive integer requiring maximum number of terms when expressed as a sum of positive n-th powers.

1, 7, 23, 79, 223, 703, 2175, 6399, 19455, 58367, 176127, 528383, 1589247, 4767743, 14319615, 42991615, 129105919, 387186687, 1161822207, 3486515199, 10458497023, 31377588223, 94136958975, 282427654143, 847282962431, 2541815332863

Offset

1,2

Comments

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).

References

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

Links

T. D. Noe, Table of n, a(n) for n = 1..200

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, exercise 7.1.1. p. 277.

Eric Weisstein's World of Mathematics, Waring's Problem.

Formula

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

Example

a(3) = 23 = 16 +  7 = 2*(2^3) +  7*(1^3) is a sum of 9 cubes;
a(4) = 79 = 64 + 15 = 4*(2^4) + 15*(1^4) is a sum of 19 biquadrates.

Maple

A018886 := proc(n)
2^n*floor((3/2)^n)-1
end proc: # R. J. Mathar, May 07 2015

Mathematica

a[n_]:=-1+2^n*Floor[(3/2)^n]
a[Range[1, 20]] (* Julien Kluge, Jul 21 2016 *)

Prog

(Python)
def a(n): return (3**n//2**n-1)*2**n + (2**n-1)
print([a(n) for n in range(1, 27)]) # Michael S. Branicky, Dec 17 2021
(Python)
def A018886(n): return (3**n&-(1<<n))-1 # Chai Wah Wu, Jun 25 2024

Crossrefs

Cf. A079611, A185187.

Cf. A000079, A002379.

Sequence in context: A356684 A048539 A240526 * A145842 A086908 A093069

Adjacent sequences: A018883 A018884 A018885 * A018887 A018888 A018889

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved