A007412 The noncubes: a(n) = n + floor((n + floor(n^(1/3)))^(1/3)).

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Seems to be numbers k for which the order of the torsion subgroup t of the elliptic curve y^2 = x^3 - k is t=1. - Artur Jasinski, Jun 30 2010

A010057(a(n)) = 0. - Reinhard Zumkeller, Oct 22 2011

References

• J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 27911
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• A. J. dos Reis and D. M. Silberger, Generating nonpowers by formula, Math. Mag., 63 (1990), 53-55.
• J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
• Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
• R. D. Nelson, Sequences which omit powers, The Mathematical Gazette, Number 461, 1988, pages 208-211.

Crossrefs

Cf. A000578 (complement), A000037 (nonsquares).

Programs

Haskell

a007412 n = n + a048766 (n + a048766 n)  -- Reinhard Zumkeller, Oct 22 2011

Mathematica

With[{upto=58},Complement[Range[upto],Range[Ceiling[Power[upto, (3)^-1]]]^3]] (* Harvey P. Dale, Nov 09 2011 *)
A007412Q = ! IntegerQ[#~Surd~3] &; Select[Range[57], A007412Q] (* JungHwan Min, Mar 27 2017 *)

PARI

lista(nn) = for (n=1, nn, if (! ispower(n, 3), print1(n, ", "))); \\ Michel Marcus, May 24 2015

PARI

list(lim)=my(v=List(),s=sqrtnint(lim\=1,3),k3,k13=1); for(k=1,s, k3=k13; k13=(k+1)^3; for(n=k3+1,k13-1, listput(v,n))); for(n=s^3+1,lim, listput(v,n)); Vec(v) \\ Charles R Greathouse IV, Jun 13 2024

Python

from sympy import integer_nthroot
def A007412(n): return n+(k:=integer_nthroot(n,3)[0])+int(n>=(k+1)**3-k) # Chai Wah Wu, Jun 17 2024

Formula

a(n) = n + A048766(n + A048766(n)). - Reinhard Zumkeller, Oct 22 2011

a(n) = n + n^(1/3) + O(1). - Charles R Greathouse IV, Aug 08 2024