A077118 -id:A077118 - cp's OEIS frontend

A077119 a(n) = A077118(n) - n^3.

0, 0, 1, -2, 0, -4, 9, 18, 17, 0, 24, -35, 36, 12, -40, -11, 0, -13, -56, 30, -79, -45, -39, -67, 100, 0, 113, -83, -48, -53, -104, 138, -7, 163, -100, -26, 0, -28, -116, 217, 9, 248, -104, 17, 80, 79, 8, -139, 297, 0, 316, -155, 17, 119, 145, 89, -55

Offset: 0

Reinhard Zumkeller, Oct 29 2002

sign

a(n)=0 iff n = m^(6*k).
Values d=x^3-y^2 of extremal points of elliptic Mordell curves. Definition for extremal points see A200656. Each value x has only one value of distance d when coordinate x is extremal point, but for many fixed distances d, the elliptic curve has more than 1 extremal point. - Artur Jasinski, Nov 30 2011
Theorem (Artur Jasinski): If a(n)>0 then a(n)<(4n^(3/2)-1)/4 for every n. If a(n)<0 then a(n)>(-4n^(3/2)-1)/4 for every n. a(n)=0 then n is perfect square. - Artur Jasinski, Dec 08 2011

			A077118(10)=1024=32^2 is the nearest square to 10^3=1000, therefore a(10)=1024-1000=24.

Cf. A000578, A077118, A077111.

|a(n)| = A002938(n).

[Round(Sqrt(n^3))^2-n^3: n in [0..60]]; // Vincenzo Librandi, Mar 24 2015

A077119 := proc(n)
    (round( sqrt(n^3) ))^2-n^3 ;
end proc: # R. J. Mathar, Jan 18 2021

Table[Round[Sqrt[x^3]]^2 - x^3, {x, 0, 100}]  (* Artur Jasinski, Nov 30 2011 *)

from math import isqrt
def A077119(n): return ((m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1))**2-k # Chai Wah Wu, Jul 29 2022

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

A077116 n^3 - A065733(n).

Original entry on oeis.org

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

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

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).

A002821 a(n) = nearest integer to n^(3/2).

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 19, 23, 27, 32, 36, 42, 47, 52, 58, 64, 70, 76, 83, 89, 96, 103, 110, 118, 125, 133, 140, 148, 156, 164, 173, 181, 190, 198, 207, 216, 225, 234, 244, 253, 263, 272, 282, 292, 302, 312, 322, 333, 343, 354, 364, 375, 386, 397

Offset: 0

N. J. A. Sloane

M. Boll, Tables Numériques Universelles. Dunod, Paris, 1947, p. 46.
M. Hall, Jr., The Diophantine equation x^3-y^2=k, pp. 173-198 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
A. V. Lebedev and R. M. Fedorova, A Guide to Mathematical Tables. Pergamon, Oxford, 1960, p. 17.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000093, A077118.

a002821 = round . sqrt . fromIntegral . (^ 3)
-- Reinhard Zumkeller, Jul 11 2014

A002821 := proc(n)
    round(n^(3/2)) ;
end proc:
seq(A002821(n),n=0..100) ;

t[n_]:=Module[{flt=Floor[n],cet=Ceiling[n]},If[n-fltHarvey P. Dale, May 12 2011 *)

from math import isqrt
def A002821(n): return (m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1) # Chai Wah Wu, Jul 30 2022

a(n) = floor(n^(3/2) + 1/2). - Ridouane Oudra, Nov 13 2019

a(n) = sqrt(A077118(n)). - Chai Wah Wu, Jul 30 2022

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.

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

A077120 n^3 times nearest integer square to n^3.

Original entry on oeis.org

0, 1, 72, 675, 4096, 15125, 48600, 123823, 270848, 531441, 1024000, 1724976, 3048192, 4853173, 7419776, 11353500, 16777216, 24073700, 33685632, 47251651, 63368000, 85349376, 112964632, 147220700, 192485376, 244140625

Offset: 0

Reinhard Zumkeller, Oct 29 2002

nonn

n3ts[n_]:=Module[{n3=n^3,a,b,sr},sr=Sqrt[n3];a=(Floor[sr])^2;b=(Ceiling[ sr])^2;If[n3-aHarvey P. Dale, Nov 27 2011 *)

a(n) = A000578(n)*A077118(n).

cp's OEIS Frontend

A077119 a(n) = A077118(n) - n^3.

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A077116 n^3 - A065733(n).

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A065733 Largest square <= n^3.

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A077110 Nearest integer cube to n^2.

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A002821 a(n) = nearest integer to n^(3/2).

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

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A077115 Least integer square >= n^3.

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