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A094784 Numbers that are neither squares nor cubes.

%I A094784 #38 Jun 05 2025 20:19:38
%S A094784 2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,28,29,30,31,
%T A094784 32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,
%U A094784 57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,82
%N A094784 Numbers that are neither squares nor cubes.
%C A094784 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
%C A094784 Intersection of A000037 and A007412: (1-A010052(a(n)))*(1-A010057(a(n)))=1. - _Reinhard Zumkeller_, Jul 13 2010
%D A094784 R. Honsberger, Mathematical Chestnuts from Around the World, MAA, 2001; see p. 168.
%H A094784 Reinhard Zumkeller, <a href="/A094784/b094784.txt">Table of n, a(n) for n = 1..10000</a>
%H A094784 J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%t A094784 Select[Range[100], !IntegerQ[#^(1/2)] && !IntegerQ[#^(1/3)]&] (* _Jean-François Alcover_, Feb 07 2020 *)
%o A094784 (PARI) is(n)=!issquare(n) && !ispower(n,3) \\ _Charles R Greathouse IV_, Oct 19 2015
%o A094784 (Haskell)
%o A094784 a094784 n = a094784_list !! (n-1)
%o A094784 a094784_list = [x | x <- [0..], a010052 x == 0, a010057 x == 0]
%o A094784 -- _Reinhard Zumkeller_, Jan 31 2012
%o A094784 (Magma) [n: n in [0..90] | not IsSquare(n) and not IsPower(n,3)]; // _Bruno Berselli_, Feb 22 2016
%o A094784 (Python)
%o A094784 from math import isqrt
%o A094784 from sympy import integer_nthroot
%o A094784 def A094784(n):
%o A094784     def f(x): return n+isqrt(x)+integer_nthroot(x,3)[0]-integer_nthroot(x,6)[0]
%o A094784     m, k = n, f(n)
%o A094784     while m != k: m, k = k, f(k)
%o A094784     return m # _Chai Wah Wu_, Jun 05 2025
%Y A094784 A005117 (squarefree numbers) intersect A004709 (cubefree numbers) is A005117; A005117 union A004709 is A004709.
%K A094784 nonn,easy
%O A094784 1,1
%A A094784 _Cino Hilliard_, Jun 10 2004
%E A094784 Definition corrected by _Rick L. Shepherd_, Aug 11 2004
%E A094784 Comment corrected by _Reinhard Zumkeller_, Jul 18 2010