A077116 -id:A077116 - cp's OEIS frontend

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

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 *)

A070929 Smallest integer >= 0 of the form x^2 - n^3.

Original entry on oeis.org

0, 0, 1, 9, 0, 19, 9, 18, 17, 0, 24, 38, 36, 12, 65, 106, 0, 128, 97, 30, 100, 148, 168, 154, 100, 0, 113, 198, 249, 260, 225, 138, 356, 163, 297, 389, 0, 423, 353, 217, 9, 248, 441, 17, 80, 79, 8, 506, 297, 0, 316, 574, 17, 119, 145, 89, 784, 568, 252, 737, 225, 548

Offset: 0

Benoit Cloitre, May 20 2002

a(n)=0 iff n is a square.

			A077115(10) = 1024 = 32^2 is the least square >= 10^3 = 1000, therefore a(10) = 1024 - 1000 = 24.

Cf. A000578, A077116, A077118, A077119, A070923, A068527.

f[n_]=Ceiling[n^(3/2)]^2-n^3;
t1=Table[f[n], {n, 1, 90}]; t1 (* Clark Kimberling, Jan 30 2011 *)

for(n=1,100,print1(ceil(n^(3/2))^2-n^3,","))

a(n) = ceiling(n^(3/2))^2 - n^3 = A077115(n) - n^3.

A154333 Difference between n^3 and the next smaller square.

Original entry on oeis.org

1, 4, 2, 15, 4, 20, 19, 28, 53, 39, 35, 47, 81, 40, 11, 127, 13, 56, 135, 79, 45, 39, 67, 135, 249, 152, 83, 48, 53, 104, 207, 7, 216, 100, 26, 431, 28, 116, 270, 496, 277, 104, 546, 503, 524, 615, 139, 368, 685, 391, 155, 732, 652, 648, 726, 55, 293, 631, 170, 704, 405

Offset: 1

M. F. Hasler, Jan 07 2009

nonn

The sequence A077116(n) = n^3-[sqrt(n^3)]^2 satisfies A077116(n)=0 <=> n^3 is a square <=> n is a square. It differs from the present sequence (which is always positive) only in these indices, where a(k^2)=2k^3-1.

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

Cf. A087285 (range of this sequence, excluding the initial term 1).

A154333 := proc(n)
    A071797(n^3) ;
end proc: # R. J. Mathar, May 29 2016

nss[n_]:=Module[{n3=n^3,s},s=Floor[Sqrt[n3]]^2;If[s==n3,s=(Sqrt[s]- 1)^2, s]]; Table[n^3-nss[n],{n,70}] (* Harvey P. Dale, Jan 19 2017 *)

A154333(n) = n^3-sqrtint(n^3-1)^2
a154333 = vector(90,n,n^3-sqrtint(n^3-1)^2)

a(n) = n^3 - [sqrt(n^3 - 1)]^2 = A000578(n) - A048760(n^3-1). a(k^2) = 2 k^3 - 1.

a(n) = A071797(n^3). - R. J. Mathar, May 29 2016

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.

A176582 n^2+Largest cube, (Largest cube <= n^2).

Original entry on oeis.org

0, 2, 5, 17, 24, 33, 63, 76, 128, 145, 164, 185, 269, 294, 321, 441, 472, 505, 540, 704, 743, 784, 827, 1041, 1088, 1137, 1188, 1458, 1513, 1570, 1629, 1690, 2024, 2089, 2156, 2225, 2296, 2700, 2775, 2852, 2931, 3012, 3492, 3577, 3664, 3753, 3844, 4406

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

9+8=17, 16+8=24, 25+8=33, 36+27=63,..

Cf. A065733, A075847, A077116, A176580, A176581.

r[n_]:=n^2;f[n_]:=r[n]+Floor[r[n]^(1/3)]^3;Table[f[n],{n,0,5!}]

A176583 n^2+Smallest cube, (Smallest cube >= n^2).

Original entry on oeis.org

0, 2, 12, 36, 43, 52, 100, 113, 128, 206, 225, 246, 360, 385, 412, 568, 599, 632, 667, 873, 912, 953, 996, 1258, 1305, 1354, 1405, 1458, 1784, 1841, 1900, 1961, 2355, 2420, 2487, 2556, 2627, 3097, 3172, 3249, 3328, 3409, 3961, 4046, 4133, 4222, 4313, 4953

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

4+8=12, 9+27=36, 16+27=43,..

Cf. A065733, A075847, A077116, A176580, A176581, A176582

r[n_]:=n^2;f[n_]:=r[n]+Ceiling[r[n]^(1/3)]^3;Table[f[n],{n,0,5!}]
Table[n^2+Ceiling[Surd[n^2,3]]^3,{n,0,50}] (* Harvey P. Dale, Jun 04 2016 *)

A176584 Primes of the form p = k^3 + (largest square <= k^3).

Original entry on oeis.org

2, 194057, 601067, 1607173, 3283993, 3630257, 3721363, 4918999, 5255713, 5848307, 6749569, 9999907, 17992913, 20432729, 23393483, 34746203, 44845993, 73843813, 84277273, 107165033, 109735649, 120469409, 125138873, 130898927

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

Consider k = 4 * x^2 + 1, where the largest square <= k^3 is (8 * x^3 + 3 * x)^2. Bunyakowsky's conjecture implies there are infinitely many primes of the form (4 * x^2 + 1)^3 + (8 * x^3 + 3 * x)^2, and so that the sequence is infinite. The first term of this form is 93389778901 = a(165), corresponding to x = 30. - Robert Israel, Jan 10 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Primes in A176580.

Cf. A065733, A075847, A077116, A176581, A176582, A176583.

g:= n -> n + floor(sqrt(n))^2:
select(isprime,map(g,[seq(i^3,i=1..1000)])); # Robert Israel, Jan 10 2024

r[n_]:=n^3;f[n_]:=r[n]+Floor[Sqrt[r[n]]]^2;Select[Table[f[n],{n,0,6!}],PrimeQ[ # ]&]

select(isprime, vector(1000, n, n^3+sqrtint(n^3)^2)) \\ Michel Marcus, Jan 10 2024

A176585 Primes of the form n^3+Smallest square, (Smallest square >= n^3).

Original entry on oeis.org

2, 17, 269, 281233, 1770217, 1826609, 2520841, 3907529, 7595017, 8665471, 9828089, 11280377, 12259063, 17235221, 27654961, 54008809, 70583033, 75196799, 85018949, 87240233, 106316057, 111499057, 168061561, 176255669, 201105409

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

8+9=17, 5^3+12^2=269,..

Cf. A065733, A075847, A077116, A176580, A176581, A176582, A176583, A176584

r[n_]:=n^3;f[n_]:=r[n]+Ceiling[Sqrt[r[n]]]^2;Select[Table[f[n],{n,0,6!}],PrimeQ[ # ]&]
ssn3[n_]:=n^3+(Ceiling[Sqrt[n^3]])^2; Select[Array[ssn3,500],PrimeQ] (* Harvey P. Dale, Jun 23 2017 *)

cp's OEIS Frontend

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

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A070929 Smallest integer >= 0 of the form x^2 - n^3.

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A154333 Difference between n^3 and the next smaller square.

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

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Maple

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

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

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A176582 n^2+Largest cube, (Largest cube <= n^2).

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Mathematica

A176583 n^2+Smallest cube, (Smallest cube >= n^2).

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Mathematica

A176584 Primes of the form p = k^3 + (largest square <= k^3).

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