A077121 -id:A077121 - cp's OEIS frontend

A167222 Irregular triangle read by rows: T(n,k) = n^3 - k^2 with 0 <= k <= A077121(n).

0, 1, 0, 8, 7, 4, 27, 26, 23, 18, 11, 2, 64, 63, 60, 55, 48, 39, 28, 15, 0, 125, 124, 121, 116, 109, 100, 89, 76, 61, 44, 25, 4, 216, 215, 212, 207, 200, 191, 180, 167, 152, 135, 116, 95, 72, 47, 20, 343, 342, 339, 334, 327, 318, 307, 294, 279, 262, 243, 222, 199, 174

Offset: 0

Reinhard Zumkeller, Oct 31 2009

			Triangle begins:
  0;
  1, 0;
  8, 7, 4;
  27, 26, 23, 18, 11, 2;
  64, 63, 60, 55, 48, 39, 28, 15, 0;
  125, 124, 121, 116, 109, 100, 89, 76, 61, 44, 25, 4;
  216, 215, 212, 207, 200, 191, 180, 167, 152, 135, 116, 95, 72, 47, 20;
  ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Cf. A000290, A000578, A077121.

For primes see A167224.

row(n) = my(c=n^3); vector(1+sqrtint(c), i, c-(i-1)^2); \\ Michel Marcus, May 28 2024

A167224 Table of primes of the form n^3 - k^2, 0<=k<=A077121(n).

Original entry on oeis.org

7, 23, 11, 2, 109, 89, 61, 191, 167, 47, 307, 199, 19, 503, 487, 463, 431, 223, 151, 71, 53, 991, 919, 271, 1327, 1231, 1187, 547, 431, 307, 1607, 1559, 1439, 1367, 1103, 887, 503, 359, 47, 2161, 2053, 1873, 1621, 1297, 433, 2719, 2663, 2383, 1783, 1223, 1063

Offset: 1

Reinhard Zumkeller, Oct 31 2009

Primes in A167222;

A161681 is the range of this table.

			7;23,11,2;;109,89,61;191,167,47;307,199,19;503,487,... .

R. Zumkeller, Table of n, a(n) for n = 1..10000
R. Zumkeller, Some Examples

A167223 Number of primes of the form n^3 - k^2, 0<=k<=A077121(n).

Original entry on oeis.org

0, 0, 1, 3, 0, 3, 3, 3, 7, 1, 3, 6, 9, 6, 7, 11, 1, 20, 13, 5, 19, 11, 8, 15, 15, 0, 17, 22, 11, 22, 16, 7, 39, 28, 8, 29, 1, 12, 31, 22, 16, 46, 33, 13, 32, 30, 13, 58, 43, 0, 47, 22, 28, 49, 39, 20, 47, 51, 18, 44, 32, 21, 84, 63, 0, 70, 38, 28, 113, 45, 23, 43, 66, 46, 52, 63, 28, 78

Offset: 0

Reinhard Zumkeller, Oct 31 2009

nonn

Number of terms per row in the table of A167224;

a(n) <= A077121(n).

			a(3) = #{27-4, 27-16, 27-25} = #{23, 11, 2} = 3;
a(4) = #{} = 0;
a(5) = #{125-16, 125-36, 125-64} = #{109, 89, 61} = 3.

R. Zumkeller, Table of n, a(n) for n = 0..500

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

Original entry on oeis.org

0, 1, 2, 5, 8, 11, 14, 18, 22, 27, 31, 36, 41, 46, 52, 58, 64, 70, 76, 82, 89, 96, 103, 110, 117, 125, 132, 140, 148, 156, 164, 172, 181, 189, 198, 207, 216, 225, 234, 243, 252, 262, 272, 281, 291, 301, 311, 322, 332, 343, 353, 364, 374, 385, 396, 407, 419, 430

Offset: 0

N. J. A. Sloane

B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Christian G. Bower, Table of n, a(n) for n=0..500
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

Integer part of square root of n^k: A000196 (k=1), this sequence (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), A155016 (k=13), A155018 (k=15), A155019 (k=17).
Cf. A002821.
Cf. A185549.

a000093 = a000196 . a000578  -- Reinhard Zumkeller, Jul 11 2014

Digits := 100: A000093 := n->floor(evalf(n^(3/2)));

Mathematica
```
Table[ Floor[ Sqrt[n^3]], {n, 0, 60}]
```
PARI
```
a(n)=if(n<0,0,sqrtint(n^3))
```

from math import isqrt
def A000093(n): return isqrt(n**3) # Chai Wah Wu, Sep 08 2024

a(n) = A077121(n) - 1. [Reinhard Zumkeller, Oct 31 2009]
a(n) = floor(n*sqrt(n)). [Arkadiusz Wesolowski, Jun 01 2011]
a(n) = A000196(A000578(n)) = A074704(n)+n*A000196(n). [Reinhard Zumkeller, Jun 27 2011]

More terms from James Sellers, May 04 2000

A065733 Largest square <= n^3.

Original entry on oeis.org

0, 1, 4, 25, 64, 121, 196, 324, 484, 729, 961, 1296, 1681, 2116, 2704, 3364, 4096, 4900, 5776, 6724, 7921, 9216, 10609, 12100, 13689, 15625, 17424, 19600, 21904, 24336, 26896, 29584, 32761, 35721, 39204, 42849, 46656, 50625, 54756, 59049, 63504

Offset: 0

Labos Elemer, Nov 15 2001

			a(10) = 961, as 961 = 31^2 is the largest square <= 1000 = 10^3.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000093, A048760, A065730-A065741, A087285.

Cf. A077115, A077116, A077118, A077121.

a065733 n = head [x | x <- reverse [0.. n^3], a010052 x == 1] -- Reinhard Zumkeller, Oct 10 2013

Table[Floor[Sqrt[w^3]//N]^2, {w, 1, 50}]

A065733(n)=sqrtint(n^3)^2  \\ M. F. Hasler, Oct 05 2013

a(n) + A077116(n) = n^3.
a(n) = A048760(n^3).
n^3 - 2*n^(3/2) <= a(n) <= n^3. - Charles R Greathouse IV, Dec 05 2022
a(n) = A000093(n)^2. - Amiram Eldar, Jul 14 2024

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

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 *)

cp's OEIS Frontend

A167222 Irregular triangle read by rows: T(n,k) = n^3 - k^2 with 0 <= k <= A077121(n).

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A167224 Table of primes of the form n^3 - k^2, 0<=k<=A077121(n).

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A167223 Number of primes of the form n^3 - k^2, 0<=k<=A077121(n).

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

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A065733 Largest square <= n^3.

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A077113 Number of nonnegative integer cubes <= 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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