A188462 Least number of 5th powers needed to represent n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Offset

1,2

Comments

Vaughan & Wooley (1995) prove that a(n) <= 17 for large enough n; in fact it is conjectured that a(n) <= 6 for large enough n. The maximum value is a(223) = 37. - Charles R Greathouse IV, Jul 05 2017

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, Number theory for the millennium, III (Urbana, IL, 2000), 301-340, A K Peters, Natick, MA, 2002.

Robert C. Vaughan and Trevor D. Wooley, Further improvements in Waring's problem, Acta Mathematica 174:2 (1995), pp. 147-240.

Example

33 = 2^5 + 1^5 (least decomposition) hence a(33) = 2.

Mathematica

Cnt5[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 5]] == 0, k++]; k]; Array[Cnt5, 105] (* T. D. Noe, Apr 01 2011 *)

Prog

(Python)
from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A188462(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n, 5, k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

Crossrefs

Cf. A002828 (squares), A002376 (cubes), A002377 (4th powers), A374012 (6th powers).

Sequence in context: A158377 A167975 A008684 * A352530 A058321 A318892

Adjacent sequences: A188459 A188460 A188461 * A188463 A188464 A188465

Keyword

nonn

Author

Jean-François Alcover, Apr 01 2011

Status

approved