A070929 -id:A070929 - 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

A070923 a(n) is the smallest value >= 0 of the form x^3 - n^2.

Original entry on oeis.org

0, 0, 4, 18, 11, 2, 28, 15, 0, 44, 25, 4, 72, 47, 20, 118, 87, 54, 19, 151, 112, 71, 28, 200, 153, 104, 53, 0, 216, 159, 100, 39, 307, 242, 175, 106, 35, 359, 284, 207, 128, 47, 433, 348, 261, 172, 81, 535, 440, 343, 244, 143, 40, 566, 459, 350, 239, 126, 11, 615, 496

Offset: 0

Benoit Cloitre, May 20 2002

a(n) = 0 if n is a cube (i.e., n is in A000578(k)).

a(n) = A181138(n) if n is not a cube. - Zak Seidov, Mar 26 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000290, A068527, A070929, A075847, A077110, A077111, A181138.

a(n) = ceiling(n^(2/3))^3 - n^2 = A077107(n)-n^2.

a(0)=0 prepended by Alois P. Heinz, Mar 07 2022

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

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

Original entry on oeis.org

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).

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

A233422 Numbers n such that m - n^3 is a square, where m is the least square above n^3.

Original entry on oeis.org

0, 2, 3, 6, 12, 20, 24, 30, 40, 42, 56, 60, 68, 75, 78, 84, 87, 120, 126, 160, 180, 248, 264, 270, 273, 308, 312, 318, 330, 336, 351, 360, 396, 564, 570, 588, 615, 620, 630, 635, 720, 738, 780, 840, 912, 1008, 1016, 1032, 1284, 1308, 1320, 1334, 1344, 1404, 1540, 1617

Offset: 1

Alex Ratushnyak, Dec 09 2013

nonn

Numbers n such that A070929(n) is a nonzero square.
The sequence of cubes a(n)^3 begins: 0, 8, 27, 216, 1728, 8000, 13824, 27000, 64000, 74088, 175616, 216000, 314432, ...
The sequence of m's begins: 1, 9, 36, 225, 1764, 8100, 13924, 27225, 64009, 74529, 176400, 216225, 314721, ...
The sequence of square roots of these m's begins: 1, 3, 6, 15, 42, 90, 118, 165, 253, 273, 420, 465, 561, 650, 689, 770, 812, ...
The sequence of squares m-n^3 begins: 1, 1, 9, 9, 36, 100, 100, 225, 9, 441, 784, 225, 289, 625, 169, 196, 841, ...
The sequence of their square roots begins: 1, 1, 3, 3, 6, 10, 10, 15, 3, 21, 28, 15, 17, 25, 13, 14, 29, 35, 43, 24, ... (note the first 12 terms are triangular numbers, A000217).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A070929, A077115, A154101, A233400, A233401.

fQ[n_]:=Module[{c=n^3,m},m=(Floor[Sqrt[c]]+1)^2;IntegerQ[Sqrt[m-c]]];Select[Range[0,1650],fQ] (* Harvey P. Dale, Jan 03 2024 *)

is(n)=issquare((sqrtint(n=n^3)+1)^2-n) \\ Charles R Greathouse IV, Dec 09 2013

from math import isqrt
def isSquare(a):
  sr = isqrt(a)
  return (a==sr*sr)
for n in range(77777):
  n3 = n*n*n
  a = isqrt(n3)+1
  if isSquare(a*a-n3):  print(n, end=', ')

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A077116 n^3 - A065733(n).

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A070923 a(n) is the smallest value >= 0 of the form x^3 - n^2.

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A077118 Nearest integer square to n^3.

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

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A077115 Least integer square >= n^3.

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A233422 Numbers n such that m - n^3 is a square, where m is the least square above n^3.

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