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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Examples

			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.
		

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.

Crossrefs

Cf. A079611, A185187.

Cf. A000079, A002379.

Programs

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 *)

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<Chai Wah Wu, Jun 25 2024

Formula

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