A007412 The noncubes: a(n) = n + floor((n + floor(n^(1/3)))^(1/3)). (Formerly M0493)

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

1,1

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.

Formula

a(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 *)

Prog

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

Crossrefs

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

Sequence in context: A102352 A248182 A219227 * A270420 A336592 A335275

Adjacent sequences: A007409 A007410 A007411 * A007413 A007414 A007415

Keyword

nonn,easy,nice,changed

Author

N. J. A. Sloane

Status

approved