A180499 - OEIS

%I #26 Aug 13 2024 02:24:46

%S 2,10,30,68,130,222,350,521,739,1011,1343,1741,2211,2759,3392,4114,

%T 4932,5852,6880,8022,9284,10673,12193,13852,15654,17606,19714,21985,

%U 24423,27035,29827,32805,35975,39343

%N n^3 + n-th cubefree number.

%C First differs from n^3 + n (A034262) at n=8 because 8 is the first positive integer which is not cubefree.

%C What cubes appear in this sequence?

%C No cubes appear in this sequence: the n-th cubefree number is asymptotically zeta(3)*n, putting members of this sequence strictly between n^3 and (n+1)^3. (Lacking effective error bounds this actually only shows that there are finitely many.) - _Charles R Greathouse IV_, Jan 21 2011

%H Charles R Greathouse IV, <a href="/A180499/b180499.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n^3 + A004709(n) = A000578(n) + A004709(n).

%e a(8) = 8^3 + 8th number that is not divisible by any cube > 1 = 8^3 + 9 = 521.

%t #[[1]]+#[[2]]^3&/@Module[{cf=Select[Range[50],Max[FactorInteger[#][[All,2]]] < 3&]},Thread[{cf,Range[Length[cf]]}]] (* _Harvey P. Dale_, Jun 28 2020 *)

%o (Python)

%o from sympy import mobius, integer_nthroot

%o def A180499(n):

%o     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return n**3+m # _Chai Wah Wu_, Aug 12 2024

%Y Cf. A000578, A004709, A034262, A161203.

%K nonn,easy

%O 1,1

%A _Jonathan Vos Post_, Jan 20 2011