A077111 -id:A077111 - cp's OEIS frontend

A075847 Difference between n^2 and the largest cube <= n^2.

0, 0, 3, 1, 8, 17, 9, 22, 0, 17, 36, 57, 19, 44, 71, 9, 40, 73, 108, 18, 57, 98, 141, 17, 64, 113, 164, 0, 55, 112, 171, 232, 24, 89, 156, 225, 296, 38, 113, 190, 269, 350, 36, 121, 208, 297, 388, 12, 107, 204, 303, 404, 507, 65, 172, 281, 392, 505, 620, 106, 225, 346

Offset: 0

Zak Seidov and Reinhard Zumkeller, Oct 15 2002

a(n) = n^2 - A077106(n).
a(n) = 0 iff n = m^(6*k).
a(n) = 0 when n is a cube. See A070923.

			a(4)=8 because 4^2 - 2^3 = 8; a(9)=17 because 9^2 - 4^3 = 17.
A077106(20) = 343 = 7^3 is the largest cube <= 20^2 = 400, therefore a(20) = 400 - 343 = 57.

Zak Seidov, Table of n, a(n) for n = 0..2000

Cf. A000290, A070923, A077110, A077111, A077116.

Cf. A070923.

Table[a = n^2; a - Floor[a^(1/3)]^3, {n, 0, 300}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

Edited by N. J. A. Sloane at the suggestion of Zak Seidov, Oct 30 2008

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

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).

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).

A073072 Minimum value of abs(n^2-x^3) x>0.

Original entry on oeis.org

0, 3, 1, 8, 2, 9, 15, 0, 17, 25, 4, 19, 44, 20, 9, 40, 54, 19, 18, 57, 71, 28, 17, 64, 104, 53, 0, 55, 112, 100, 39, 24, 89, 156, 106, 35, 38, 113, 190, 128, 47, 36, 121, 208, 172, 81, 12, 107, 204, 244, 143, 40, 65, 172, 281, 239, 126, 11, 106, 225, 346, 252, 127, 0, 129

Offset: 1

Benoit Cloitre, Aug 17 2002

Cf. A002938.

Table[Min[n^2-Floor[(n^2)^(1/3)]^3,(Floor[(n^2)^(1/3)]+1)^3-n^2],{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

a(n)=vecmin(vector(ceil(n^(2/3)),i,abs(n^2-i^3)))

a(n^3) = 0.

a(n) = abs(A077111(n)). - R. J. Mathar, May 01 2008

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A075847 Difference between n^2 and the largest cube <= n^2.

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

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

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A077110 Nearest integer cube to n^2.

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A073072 Minimum value of abs(n^2-x^3) x>0.

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