A077113 -id:A077113 - cp's OEIS frontend

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

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

A077121 Number of integer squares <= n^3.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 15, 19, 23, 28, 32, 37, 42, 47, 53, 59, 65, 71, 77, 83, 90, 97, 104, 111, 118, 126, 133, 141, 149, 157, 165, 173, 182, 190, 199, 208, 217, 226, 235, 244, 253, 263, 273, 282, 292, 302, 312, 323, 333, 344, 354, 365, 375, 386, 397, 408, 420

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) = number of terms in n-th row of A167222. - Reinhard Zumkeller, Oct 31 2009

			Squares <= 3^3 = 27: 0, 1, 4, 9, 16 and 25, hence a(3) = 6.

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

Cf. A000093, A065733, A077113, A167222, A167223.

a[n_] := Floor[Sqrt[n^3]] + 1; Array[a, 100, 0] (* Amiram Eldar, Apr 06 2025 *)

from math import isqrt
def A077121(n): return isqrt(n**3)+1 # Chai Wah Wu, Sep 08 2024

a(n) = floor(n^(3/2))+1 = A000093(n) + 1.

A077106 Largest integer cube <= n^2.

Original entry on oeis.org

0, 1, 1, 8, 8, 8, 27, 27, 64, 64, 64, 64, 125, 125, 125, 216, 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

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(20) = 343, as 343 = 7^3 is the largest cube <= 400 = 20^2.

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

Cf. A000578, A000290, A065733, A075847, A077107, A077110, A077113.

a[n_] := Floor[Surd[n^2, 3]]^3; Array[a, 100, 0] (* Amiram Eldar, Apr 05 2025 *)

a(n) + A075847(n) = n^2.

a(n) = A100196(n)^3. - Amiram Eldar, Apr 06 2025

A139753 a(2n) = next cube after a(2n-1), a(2n+1) = next square after a(2n).

Original entry on oeis.org

1, 8, 9, 27, 36, 64, 81, 125, 144, 216, 225, 343, 361, 512, 529, 729, 784, 1000, 1024, 1331, 1369, 1728, 1764, 2197, 2209, 2744, 2809, 3375, 3481, 4096, 4225, 4913, 5041, 5832, 5929, 6859, 6889, 8000, 8100, 9261, 9409, 10648, 10816, 12167, 12321, 13824

Offset: 1

Zak Seidov, May 19 2008

nonn

Terms with even indices are cubes n^3 with n=2,3,... (all integers >1), while terms with odd indices are square m^3 with m=1,3,6,9,12,15,19,23,28,32,37,42,47,53,59,65,71,77,83,90,97,104,111,118,126,133,141,149,157,165,173,182,190,199,208,217,226,235,244,253,263,273,282,292,302,312,323,333,344,354,365,375,386,397,408,420,431,442,454,465,477,489,501,513,525,537,549,561,574,586,599,611,624,637,650,663,676,689,703,716,730,743,757,770,784,798,812,826,840,854,869,883,897,912,926,941,956,971,986,1001; cf. A077121 Number of integer squares <= n^3.

			a(1)=1 considered as square,
a(2)=8 = least cube >a(1);
a(3)=9 = least square >a(2),
a(4)=27 = least cube >a(3),
a(5)=36 = least square >a(4),
a(6)=64 = least cube >a(5),
a(7)=81 = least square >a(6),
a(8)=125 = least cube >a(7).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A077113, A077121.

nxt[{n_,a_}]:={n+1,If[OddQ[n],(Floor[Surd[a,3]]+1)^3,(Floor[Sqrt[a]]+1)^2]}; NestList[nxt,{1,1},50][[All,2]] (* Harvey P. Dale, Feb 09 2022 *)

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

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A077121 Number of integer squares <= n^3.

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A077106 Largest integer cube <= n^2.

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A139753 a(2n) = next cube after a(2n-1), a(2n+1) = next square after a(2n).

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