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Sum of first n noncubes.
2

%I #31 Sep 03 2024 08:26:27

%S 2,5,9,14,20,27,36,46,57,69,82,96,111,127,144,162,181,201,222,244,267,

%T 291,316,342,370,399,429,460,492,525,559,594,630,667,705,744,784,825,

%U 867,910,954,999,1045,1092,1140,1189,1239,1290,1342,1395,1449,1504,1560

%N Sum of first n noncubes.

%F a(n) = Sum_{i=1..n} A007412(i).

%F a(n) = A000217(A007412(n)) - Sum_{i=1..floor((A007412(n)^(1/3)))} i^3.

%F a(n) = A000217(A007412(n)) - A000217(floor(A007412(n)^(1/3)))^2.

%F Let R = A007412(n) and S = floor(R^(1/3)); then a(n) = (R*(R+1))/2 - ((S*(S+1))/2)^2. - _Gerald Hillier_, Dec 21 2008

%e a(6) = 2 + 3 + 4 + 5 + 6 + 7 = 27.

%e a(7) = 2 + 3 + 4 + 5 + 6 + 7 + 9 = 36.

%t Accumulate[With[{no=60},Complement[Range[no],Range[Floor[Power[no, (3)^-1]]]^3]]] (* _Harvey P. Dale_, Feb 14 2011 *)

%o (Python)

%o from sympy import integer_nthroot

%o def A109470(n): return ((m:=n+(k:=integer_nthroot(n,3)[0])+int(n>=(k+1)**3-k))*(m+1)>>1)-((r:=integer_nthroot(m,3)[0])*(r+1)>>1)**2 # _Chai Wah Wu_, Jun 17 2024

%o (PARI) a(n) = sum(i=1, n, i + sqrtnint(i + sqrtnint(i, 3), 3)); \\ _Michel Marcus_, Jun 20 2024

%Y Cf. A000537, A007412, A048766, A064524, A086849.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Aug 28 2005