A109470 Sum of first n noncubes.

2, 5, 9, 14, 20, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 201, 222, 244, 267, 291, 316, 342, 370, 399, 429, 460, 492, 525, 559, 594, 630, 667, 705, 744, 784, 825, 867, 910, 954, 999, 1045, 1092, 1140, 1189, 1239, 1290, 1342, 1395, 1449, 1504, 1560

Offset: 1

Author

Jonathan Vos Post, Aug 28 2005

Keywords

• easy
• nonn

Examples

			a(6) = 2 + 3 + 4 + 5 + 6 + 7 = 27.
a(7) = 2 + 3 + 4 + 5 + 6 + 7 + 9 = 36.
		

Crossrefs

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

Programs

Mathematica

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

PARI

a(n) = sum(i=1, n, i + sqrtnint(i + sqrtnint(i, 3), 3)); \\ Michel Marcus, Jun 20 2024

Python

from sympy import integer_nthroot
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

Formula

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

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

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

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