A065733 -id:A065733 - cp's OEIS frontend

A077116 n^3 - A065733(n).

0, 0, 4, 2, 0, 4, 20, 19, 28, 0, 39, 35, 47, 81, 40, 11, 0, 13, 56, 135, 79, 45, 39, 67, 135, 0, 152, 83, 48, 53, 104, 207, 7, 216, 100, 26, 0, 28, 116, 270, 496, 277, 104, 546, 503, 524, 615, 139, 368, 0, 391, 155, 732, 652, 648, 726, 55, 293, 631, 170, 704

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) = 0 for n = m^2. - Zak Seidov, May 11 2007

It has been asked whether some primes do not occur in this sequence. It seems indeed that primes 3, 5, 17, 23, 29, 31, 37, 41, 43, 59, 61,... do not occur, primes 2, 7, 11, 13, 19, 47, 53, 67, 79, 83,... do. For further investigations, see A087285 = the range of this sequence, and also the related sequences A229618 = range of A181138, and A165288. - M. F. Hasler, Sep 26 2013 and Oct 05 2013

			A065733(10) = 961 = 31^2 is the largest square less than or equal to 10^3 = 1000, therefore a(10) = 1000 - 961 = 39.

Cf. A000578, A070929, A077118, A077119, A075847.

A077116 := proc(n)
    A053186(n^3) ;
end proc: # R. J. Mathar, Jul 12 2016

Table[c = n^3; c - Floor[Sqrt[c]]^2, {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 02 2011 *)

A077116(n)=n^3-sqrtint(n^3)^2 \\ - M. F. Hasler, Sep 26 2013

a(n) = A154333(n) unless n is a square or, equivalently, a(n)=0. - M. F. Hasler, Oct 05 2013

a(n) = A053186(n^3). - R. J. Mathar, Jul 12 2016

A087285 Possible differences between a cube and the next smaller square.

Original entry on oeis.org

2, 4, 7, 11, 13, 15, 19, 20, 26, 28, 35, 39, 40, 45, 47, 48, 49, 53, 55, 56, 60, 63, 67, 74, 76, 79, 81, 83, 100, 104, 107, 109, 116, 127, 135, 139, 146, 147, 148, 150, 152, 155, 170, 174, 180, 184, 186, 191, 193, 200, 207, 212, 215, 216, 233, 235, 242, 244, 249

Offset: 1

Hugo Pfoertner, Sep 18 2003

nonn

Sequence and program were provided by Ralf Stephan Aug 28 2003.
Comment from David W. Wilson, Jan 05 2009: I believe there is an algorithm for solving x^3 - y^2 = k, which should have a finite number of solutions for any k. That means that we should in principle be able to compute this sequence.
Up to the initial 0 in A165288, these two sequences appear to be the same, but according to its current definition, A165288 should be the same as the (different) sequence A229618 = the range of the sequence A181138 (= least k>0 such that n^2+k is a cube): If n^2+k=y^3 is the smallest cube above n^2, then n^2 is not necessarily the largest square below y^3. E.g., 18 is in A181138 and A229618, since 9+18=27 is the least cube above 9=3^2, but 25=5^2 is the largest square below 27. - M. F. Hasler, Oct 05 2013

			a(1)=2 because the next smaller square below 3^3=27 is 5^2=25.

See under A081121.

Cf. A087286, A088017, A081121, A077116, A065733.

v=vector(200):for(n=2,10^7,t=n^3:s=sqrtint(t)^2: if(s==t,s=sqrtint(t-1)^2):tt=t-s: if(tt>0&&tt<=200&&!v[tt],v[tt]=n)):for(k=1,200,if(v[k],print1(k",")))

A077118 Nearest integer square to n^3.

Original entry on oeis.org

0, 1, 9, 25, 64, 121, 225, 361, 529, 729, 1024, 1296, 1764, 2209, 2704, 3364, 4096, 4900, 5776, 6889, 7921, 9216, 10609, 12100, 13924, 15625, 17689, 19600, 21904, 24336, 26896, 29929, 32761, 36100, 39204, 42849, 46656, 50625, 54756, 59536

Offset: 0

Reinhard Zumkeller, Oct 29 2002

nonn

			a(5)=121, as 121=11^2 is the nearest square to 125=5^3.

Cf. A000290, A000578, A002821, A077119, A077110.

Table[Round[Sqrt[n^3]]^2, {n, 0, 39}] (* Alonso del Arte, Dec 07 2011, based on Artur Jasinski's program for A077119 *)

from math import isqrt
def A077118(n): return ((m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1))**2 # Chai Wah Wu, Jul 29 2022

a(n) = if A077116(n) < A070929(n) then A065733(n) else A077115(n).

a(n) = A002821(n)^2. - Chai Wah Wu, Jul 30 2022

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.

A176580 n^3+Largest square, (Largest square <= n^3).

Original entry on oeis.org

0, 2, 12, 52, 128, 246, 412, 667, 996, 1458, 1961, 2627, 3409, 4313, 5448, 6739, 8192, 9813, 11608, 13583, 15921, 18477, 21257, 24267, 27513, 31250, 35000, 39283, 43856, 48725, 53896, 59375, 65529, 71658, 78508, 85724, 93312, 101278, 109628, 118368

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

2^3+4=12,3^3+25=52,..

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

Cf. A065733, A077116

r[n_]:=n^3;f[n_]:=r[n]+Floor[Sqrt[r[n]]]^2;Table[f[n],{n,0,5!}]
Table[n^3+Floor[Sqrt[n^3]]^2,{n,0,40}] (* Harvey P. Dale, May 15 2025 *)

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

A077115 Least integer square >= n^3.

Original entry on oeis.org

0, 1, 9, 36, 64, 144, 225, 361, 529, 729, 1024, 1369, 1764, 2209, 2809, 3481, 4096, 5041, 5929, 6889, 8100, 9409, 10816, 12321, 13924, 15625, 17689, 19881, 22201, 24649, 27225, 29929, 33124, 36100, 39601, 43264, 46656, 51076, 55225, 59536

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(10) = 1024, as 1024 = 32^2 is the least square >= 1000 = 10^3.

Cf. A000290, A000578, A048761, A065733, A077107, A077118, A185549.

[Ceiling(n^(3/2))^2: n in [0..50]]; // Vincenzo Librandi, Feb 17 2015

lis[n_]:=Module[{c=Sqrt[n^3]},If[IntegerQ[c],c^2,(Floor[c]+1)^2]]; Array[lis,40,0] (* Harvey P. Dale, Jan 22 2013 *)

a(n) - A070929(n) = n^3.
a(n) = ceiling(n^(3/2))^2. - Benoit Cloitre, Nov 01 2002
a(n) = A185549(n)^2. - Amiram Eldar, May 17 2025
a(n) = A048761(n^3). - Michel Marcus, May 17 2025

A176581 n^3+Smallest square, (Smallest square >= n^3).

Original entry on oeis.org

0, 2, 17, 63, 128, 269, 441, 704, 1041, 1458, 2024, 2700, 3492, 4406, 5553, 6856, 8192, 9954, 11761, 13748, 16100, 18670, 21464, 24488, 27748, 31250, 35265, 39564, 44153, 49038, 54225, 59720, 65892, 72037, 78905, 86139, 93312, 101729, 110097, 118855

Offset: 0

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

2^3+9=17,3^3+36=63,..

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A065733, A077116, A176580

seq(n^3 + ceil(sqrt(n^3))^2, n=0..100); # Robert Israel, Jun 18 2018

r[n_]:=n^3;f[n_]:=r[n]+Ceiling[Sqrt[r[n]]]^2;Table[f[n],{n,0,5!}]

Offset corrected by Robert Israel, Jun 18 2018

A229618 Numbers that are the distance between a square and the next larger cube.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 15, 18, 19, 20, 25, 26, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 67, 71, 72, 74, 76, 79, 81, 83, 87, 100, 104, 106, 107, 109, 112, 116, 118, 126, 127, 128, 135, 139, 143

Offset: 1

M. F. Hasler, Sep 26 2013

This is the range of the sequence A181138 (= least k>0 such that n^2+k is a cube). Note that this is not the same as A087285 = range of A077116 = difference between a cube and the next smaller square: If n^2+k = y^3 is the smallest cube above n^2, then n^2 is not necessarily the largest square below y^3, e.g., 9+18 = 27 = 3^3 is the least cube above 9 = 3^2, but 25 = 5^2 is the largest square below 27. Therefore the number 18 is in this sequence, but not in A087285.
See A077116 and A181138 and A179386 for motivations.
Apart from the leading 1, this is a subsequence of A106265, which does not require the square to be the next smaller one: For example, 23 = 27 - 4 = 3^3 - 2^2 is in A106265 but not in this sequence. A165288 is a subsequence of this one, except for the initial term.

			a(1) = 1 = 1^3-0^2 (but this is the only solution to y^3-x^2 = 1).
a(2) = 2 = 27-25 (= 3^3-5^2), and this is the only solution to y^3-x^2 = 2.
The number 3 is not in the sequence since there are no x, y > 0 such that y^3-x^2 = 3.
a(3) = 4 = 8-4 (= 2^3-2^2) = 125-121 (= 5^3-11^2); these are the only two solutions to y^3-x^2 = 4, for all x>11, the minimal positive y^3-x^2 is 7.

Cf. A087285, A087286, A088017, A081121, A081120, A077116, A065733.

A228948 Numbers n such that n^3 + k^2 = m^3 for some k>0, m>0.

Original entry on oeis.org

6, 7, 11, 23, 24, 26, 28, 31, 38, 42, 44, 47, 54, 55, 61, 63, 84, 91, 92, 95, 96, 99, 104, 110, 111, 112, 118, 119, 124, 138

Offset: 1

M. F. Hasler, Oct 05 2013

nonn

Cube root of perfect cubes in A087285 or in A229618 are in the present sequence, but this does not yield all terms, because these sequences require k^2 to be the largest square < m^3.

Numbers k such that Mordell's equation y^2 = x^3 - k^3 has more than 1 integral solution. (Note that it is necessary that x is positive.) In other words, numbers k such that Mordell's equation y^2 = x^3 - k^3 has solutions other than the trivial solution (k,0). - Jianing Song, Sep 24 2022

			6 is a term since the equation y^2 = x^3 - 6^3 has 5 solutions (6,0), (10,+-28), and (33,+-189). - _Jianing Song_, Sep 24 2022

Cf. A229618, A087285, A087286, A088017, A081121, A081120, A077116, A065733.
Cube root of A179419.
Cf. A356709, A356720. Complement of A356713.

More terms added by Jianing Song, Sep 24 2022 based on A179419.

cp's OEIS Frontend

A077116 n^3 - A065733(n).

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A087285 Possible differences between a cube and the next smaller square.

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A077118 Nearest integer square to n^3.

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A077121 Number of integer squares <= n^3.

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A176580 n^3+Largest square, (Largest square <= n^3).

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A077106 Largest integer cube <= n^2.

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A077115 Least integer square >= n^3.

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A176581 n^3+Smallest square, (Smallest square >= n^3).

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