A105209 Nearest integer to the cube root of n.

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Author

Pahikkala Jussi, Apr 13 2005

Keywords

• easy
• nonn

Examples

			a(16) = 3 because 16^(1/3) = 2.519842...
		

References

• Erwin Voellmy, Fünfstellige Logarithmen und Zahlentafeln, Orell Füssli Verlag, Zürich (1962).

Links

• Seung-Jin Bang, Problem 10212, The American Mathematical Monthly, Vol. 99, No. 4 (1992), p. 361, Nearest Integer Zeta Functions, solution to Problem 10212, ibid., Vol. 101, No. 6 (1994), pp. 579-580.

Crossrefs

Cf. A000194, A048766.

Programs

Maple

for n from 1 to 200 do printf(`%d,`,round(n^(1/3))) od: # James Sellers, Apr 21 2005

Mathematica

Round[Surd[Range[110],3]] (* Harvey P. Dale, Feb 28 2015 *)

PARI

a(n) = round(n^(1/3)); \\ Michel Marcus, Aug 19 2016

Python

from sympy import integer_nthroot
def A105209(n): return (m:=integer_nthroot(n,3)[0])+((n<<3)>=((m<<1)+1)**3) # Chai Wah Wu, Jun 06 2025

Formula

Sum_{n>=1} 1/a(n)^s = 3*zeta(s-2) + zeta(s)/4^s, for s > 3 (Seung-Jin Bang, 1992). - Amiram Eldar, Oct 31 2020

Extensions

More terms from James Sellers, Apr 21 2005