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a(1) = 1, a(n+1) = a(n) + floor(a(n)^(1/3)).
2

%I #28 Feb 14 2024 15:06:17

%S 1,2,3,4,5,6,7,8,10,12,14,16,18,20,22,24,26,28,31,34,37,40,43,46,49,

%T 52,55,58,61,64,68,72,76,80,84,88,92,96,100,104,108,112,116,120,124,

%U 128,133,138,143,148,153,158,163,168,173,178,183,188,193,198,203,208,213,218,224,230,236,242,248,254,260

%N a(1) = 1, a(n+1) = a(n) + floor(a(n)^(1/3)).

%C First 17 terms identical to A079645 (Integer part of the cube root of n divides n). Replacing cube root by square root gives A033638.

%H Robert Israel, <a href="/A113768/b113768.txt">Table of n, a(n) for n = 1..10000</a>

%F Conjecture: a(n) ~ (2/3)*n*sqrt((2/3)*n). - _José María Grau Ribas_, Feb 13 2024

%p A[1]:= 1:

%p for n from 1 to 100 do A[n+1] := A[n] + floor(A[n]^(1/3)) od:

%p seq(A[i],i=1..100); # _Robert Israel_, Jul 28 2019

%t NestList[#+Floor[Surd[#,+3]]&,1,70] (* _Harvey P. Dale_, Jan 21 2013 *)

%o (Magma) [n le 1 select 1 else Self(n-1)+Floor(Self(n-1)^(1/3)): n in [1..75]]; // _Vincenzo Librandi_, Jul 29 2019

%Y Cf. A033638, A048766, A079645.

%K nonn,easy

%O 1,2

%A _Jonathan Vos Post_, Jan 19 2006

%E Corrected and extended by _Harvey P. Dale_, Jan 21 2013