A094784 Numbers that are neither squares nor cubes.

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82

Offset: 1

Author

Cino Hilliard, Jun 10 2004

Keywords

• nonn
• easy

Comments

Numbers n for which order of torsion subgroup t of the elliptic curve y^2=x^3+n is t=1. - Artur Jasinski, Jun 30 2010

Intersection of A000037 and A007412: (1-A010052(a(n)))*(1-A010057(a(n)))=1. - Reinhard Zumkeller, Jul 13 2010

References

• R. Honsberger, Mathematical Chestnuts from Around the World, MAA, 2001; see p. 168.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• 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]

Crossrefs

A005117 (squarefree numbers) intersect A004709 (cubefree numbers) is A005117; A005117 union A004709 is A004709.

Programs

Haskell

a094784 n = a094784_list !! (n-1)
a094784_list = [x | x <- [0..], a010052 x == 0, a010057 x == 0]
-- Reinhard Zumkeller, Jan 31 2012

Magma

[n: n in [0..90] | not IsSquare(n) and not IsPower(n,3)]; // Bruno Berselli, Feb 22 2016

Mathematica

Select[Range[100], !IntegerQ[#^(1/2)] && !IntegerQ[#^(1/3)]&] (* Jean-François Alcover, Feb 07 2020 *)

PARI

is(n)=!issquare(n) && !ispower(n,3) \\ Charles R Greathouse IV, Oct 19 2015

Python

from math import isqrt
from sympy import integer_nthroot
def A094784(n):
    def f(x): return n+isqrt(x)+integer_nthroot(x,3)[0]-integer_nthroot(x,6)[0]
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jun 05 2025

Extensions

Definition corrected by Rick L. Shepherd, Aug 11 2004

Comment corrected by Reinhard Zumkeller, Jul 18 2010