A077106 -id:A077106 - cp's OEIS frontend

A075847 Difference between n^2 and the largest cube <= n^2.

0, 0, 3, 1, 8, 17, 9, 22, 0, 17, 36, 57, 19, 44, 71, 9, 40, 73, 108, 18, 57, 98, 141, 17, 64, 113, 164, 0, 55, 112, 171, 232, 24, 89, 156, 225, 296, 38, 113, 190, 269, 350, 36, 121, 208, 297, 388, 12, 107, 204, 303, 404, 507, 65, 172, 281, 392, 505, 620, 106, 225, 346

Offset: 0

Zak Seidov and Reinhard Zumkeller, Oct 15 2002

a(n) = n^2 - A077106(n).
a(n) = 0 iff n = m^(6*k).
a(n) = 0 when n is a cube. See A070923.

			a(4)=8 because 4^2 - 2^3 = 8; a(9)=17 because 9^2 - 4^3 = 17.
A077106(20) = 343 = 7^3 is the largest cube <= 20^2 = 400, therefore a(20) = 400 - 343 = 57.

Zak Seidov, Table of n, a(n) for n = 0..2000

Cf. A000290, A070923, A077110, A077111, A077116.

Cf. A070923.

Table[a = n^2; a - Floor[a^(1/3)]^3, {n, 0, 300}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

Edited by N. J. A. Sloane at the suggestion of Zak Seidov, Oct 30 2008

A100196 Number of positive integer cubes <= n^2.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18

Offset: 0

Giovanni Teofilatto, Dec 27 2004

Peter Munn, Table of n, a(n) for n = 0..10000

Cf. A000196, A054846, A062108, A077106, A077113, A100197.

Table[ Floor[n^(2/3)], {n, 0, 120}] (* Ray Chandler, Jan 09 2005 *)

a(n)=floor(sqrtn(n^2,3)) \\ Charles R Greathouse IV, Mar 21 2012; corrected by Michel Marcus, Mar 08 2020

a(n)=(n^2+.5)^(1/3)\1 \\ More careful rounding. - Charles R Greathouse IV, Mar 21 2012

from sympy import integer_nthroot
def A100196(n): return integer_nthroot(n**2,3)[0] # Chai Wah Wu, Aug 15 2025

a(n) = floor(n^(2/3)). - Charles R Greathouse IV, Mar 21 2012
From Amiram Eldar, Apr 05 2025:  (Start)
a(n) = A077113(n) - 1.
a(n) = A077106(n)^(1/3). (End)

Corrected and extended by Ray Chandler, Jan 09 2005

a(0)=0 inserted by Sean A. Irvine, Jun 22 2020

A077110 Nearest integer cube to n^2.

Original entry on oeis.org

0, 1, 1, 8, 8, 27, 27, 64, 64, 64, 125, 125, 125, 125, 216, 216, 216, 343, 343, 343, 343, 512, 512, 512, 512, 729, 729, 729, 729, 729, 1000, 1000, 1000, 1000, 1000, 1331, 1331, 1331, 1331, 1331, 1728, 1728, 1728, 1728, 1728, 2197, 2197, 2197

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(10) = 125, as 125 = 5^3 is the nearest cube to 100 = 10^2.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000578, A000290, A075847, A070923, A077106, A077107, A077111, A077118.

Cf. A002760 (Squares and cubes). - Zak Seidov, Oct 08 2015

nic[n_]:=Module[{n2=n^2,s3,c1,c2},s3=Surd[n2,3];c1=Floor[s3]^3;c2= Ceiling[ s3]^3;If[n2-c1Harvey P. Dale, Jul 05 2015 *)

from sympy import integer_nthroot
def A077110(n):
    n2 = n**2
    a = integer_nthroot(n2,3)[0]
    a2, a3 = a**3, (a+1)**3
    return a3 if a3+a2-2*n2 < 0 else a2 # Chai Wah Wu, Sep 24 2021

a(n) = if A075847(n) < A070923(n) then A077106(n) else A077107(n).

A077107 Least integer cube >= n^2.

Original entry on oeis.org

0, 1, 8, 27, 27, 27, 64, 64, 64, 125, 125, 125, 216, 216, 216, 343, 343, 343, 343, 512, 512, 512, 512, 729, 729, 729, 729, 729, 1000, 1000, 1000, 1000, 1331, 1331, 1331, 1331, 1331, 1728, 1728, 1728, 1728, 1728, 2197, 2197, 2197, 2197, 2197, 2744

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(20) = 512, as 512 = 8^3 is the least cube >= 400 = 20^2.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000578, A000290, A048763, A070923, A077106, A077115, A077110, A121536,

Table[Ceiling[Surd[n^2,3]]^3,{n,0,50}] (* Harvey P. Dale, Jan 02 2020 *)

a(n) - A070923(n) = n^2.
a(n) = A121536(n)^3. - Amiram Eldar, May 17 2025
a(n) = A048763(n^2). - Michel Marcus, May 17 2025

A077113 Number of nonnegative integer cubes <= n^2.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) is the least number m such that m^3 > n^2. - Zak Seidov, May 03 2005

			Cubes <= 10^2: 0, 1, 8, 27 and 64, hence a(10) = 5.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A026409, A026414, A054071, A077106, A077121, A100196.

Table[Floor[n^(2/3) + 1], {n, 0, 100}] (* Zak Seidov, May 03 2005 *)

from sympy import integer_nthroot
def A077113(n): return integer_nthroot(n**2,3)[0]+1 # Chai Wah Wu, Aug 15 2025

a(n) = floor(n^(2/3))+1.
a(n) = [x^(n^2)] (1/(1 - x))*Sum_{k>=0} x^(k^3). - Ilya Gutkovskiy, Apr 20 2018
a(n) = A100196(n) + 1. - Amiram Eldar, Apr 05 2025

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A075847 Difference between n^2 and the largest cube <= n^2.

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A100196 Number of positive integer cubes <= n^2.

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A077110 Nearest integer cube to n^2.

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A077107 Least integer cube >= n^2.

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A077113 Number of nonnegative integer cubes <= n^2.

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