A070923 -id:A070923 - cp's OEIS frontend

A071069 a(n) = min { A070923(k) | n^3 < k < (n+1)^3 }.

2, 4, 11, 13, 7, 28, 47, 49, 74, 76, 60, 109, 146, 148, 191, 193, 207, 207, 233, 301, 362, 364, 63, 433, 506, 212, 587, 174, 674, 368, 503, 769, 866, 766, 971, 368, 1082, 1071, 1199, 1201, 1322, 648, 1144, 1453, 1586, 535, 508, 944, 991, 1478, 2027, 2029, 215

Offset: 1

Benoit Cloitre, May 26 2002

Strong conjecture : for n>12, n^2/2n for any n. A simple application of the weak conjecture could be to determine if the equation x^3-y^2 = A (A integer) has no solution in integers. For example the equation x^3-y^2 = 5 would have no solution in integers since a(n)>5 for n>5 and from a direct calculus, A070923(k) is different from 5, k = 1^3 to 6^3.

The strong conjecture does not hold for n = 23, 26, 28, 30, 36, 42, 46, 47, 48, 49, ... - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 17 2004

Cf. A070959, A070923.

for(n=1,21,s=1; while(sum(i=n^3+1,(n+1)^3-1,sign(ceil(i^(2/3))^3-i^2-s))==(n+1)^3-1-n^3,s++); print1(s,","))

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 17 2004

A077109 Duplicate of A070923.

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

Offset: 0

dead

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

Original entry on oeis.org

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

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).

A181138 Least positive integer k such that n^2 + k is a cube.

Original entry on oeis.org

1, 7, 4, 18, 11, 2, 28, 15, 61, 44, 25, 4, 72, 47, 20, 118, 87, 54, 19, 151, 112, 71, 28, 200, 153, 104, 53, 271, 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

Offset: 0

Jason Earls, Oct 06 2010

a(n) = A070923(n) if n is not cube. Zak Seidov, Mar 26 2013

See A229618 for the range of this sequence. A179386 gives the range of b(n) = min{ a(m); m >= n }. The indices of jumps in this sequence are given in A179388 = { n | a(m)>a(n) for all m > n } = { 0, 5, 11, 181, 207, 225, 500, 524, 1586, ... }. - M. F. Hasler, Sep 26 2013

			a(11) = 4 because 11^2 + k is never a cube for k < 4, but 11^2 + 4 = 5^3. - _Bruno Berselli_, Jan 29 2013

Bruno Berselli, Table of n, a(n) for n = 0..1000 (Corrected Jan 19 2019)

Cf. A070923, A077116.

S:=[];
k:=1;
for n in [0..60] do
   while not IsPower(n^2+k,3) do
        k:=k+1;
   end while;
   Append(~S, k);
   k:=1;
end for;
S;  // Bruno Berselli, Jan 29 2013

Table[(1 + Floor[n^(2/3)])^3 - n^2, {n, 100}] (* Zak Seidov, Mar 26 2013 *)

A181138(n)=(sqrtnint(n^2,3)+1)^3-n^2 \\ Charles R Greathouse IV, Mar 26 2013

a(n) << n^(4/3). - Charles R Greathouse IV, Mar 26 2013

Extended to a(0)=1 by M. F. Hasler, Sep 26 2013

A077107 Least integer cube >= n^2.

Original entry on oeis.org

0, 1, 8, 27, 27, 27, 64, 64, 64, 125, 125, 125, 216, 216, 216, 343, 343, 343, 343, 512, 512, 512, 512, 729, 729, 729, 729, 729, 1000, 1000, 1000, 1000, 1331, 1331, 1331, 1331, 1331, 1728, 1728, 1728, 1728, 1728, 2197, 2197, 2197, 2197, 2197, 2744

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(20) = 512, as 512 = 8^3 is the least cube >= 400 = 20^2.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000578, A000290, A048763, A070923, A077106, A077115, A077110, A121536,

Table[Ceiling[Surd[n^2,3]]^3,{n,0,50}] (* Harvey P. Dale, Jan 02 2020 *)

a(n) - A070923(n) = n^2.
a(n) = A121536(n)^3. - Amiram Eldar, May 17 2025
a(n) = A048763(n^2). - Michel Marcus, May 17 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.

A077111 a(n) = A077110(n) - n^2.

Original entry on oeis.org

0, 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

Offset: 0

Reinhard Zumkeller, Oct 29 2002

sign

a(n)=0 iff n = m^(6*k).

			A077110(20)=343=7^3 is the nearest cube to 20^2=400, therefore a(20)=343-400=-57.

Cf. A000290, A077110, A077119.

a(n) = if A075847(n) < A070923(n) then -A075847(n) else A070923(n).

A121536 Smallest m such that m^3 >= n^2.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19

Offset: 1

Zak Seidov, Aug 05 2006

Cf. A070923, A077107, A100196.

Mathematica
```
Table[Ceiling[n^(2/3)],{n,1,100}]
```

a(n) = ceiling(n^(2/3)).
a(n) = (A070923(n)+n^2)^(1/3).
If n is a cube a(n) = A100196(n), otherwise a(n) = A100196(n)+1.
a(n) = A077107(n)^(1/3). - Amiram Eldar, May 17 2025

Offset changed to 0 and a(0) prepended by Amiram Eldar, May 17 2025

A070959 First minimum value > 0 of the form x^3-k^2 when k > n^3.

Original entry on oeis.org

4, 4, 39, 13, 152, 28, 391, 49, 804, 76, 1439, 109, 2344, 148, 3567, 193, 5156, 244, 7159, 301, 9624, 364, 12599, 433, 16132, 508, 20271, 589, 25064, 676, 30559, 769, 36804, 868, 43847, 973, 51736, 1084, 60519, 1201, 70244, 1324, 80959, 1453, 92712

Offset: 1

Benoit Cloitre, May 25 2002

			Let n=2 then n^3=8 and A070923(9)= 44, A070923(10)=25, A070923(11)=4, A070923(12)=72 so the first minimum is 4, hence a(2)=4

Cf. A070923.

for(n=1,100,s=n^3+1; while(ceil(s^(2/3))^3-s^2>ceil((s+1)^(2/3))^3-(s+1)^2,s++); print1(ceil(s^(2/3))^3-s^2,","))

Let k be the smallest integer>n^3 such that A070923(k-1)> A070923(k) and such that A070923(k) < A070923(k+1), then a(n)= A070923(k); for n>=1 a(2n-1) = 8n^3-9n^2+6n-1, a(2n)=3n^2+1.
From Chai Wah Wu, Jul 27 2020: (Start)
a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n > 8.
G.f.: x*(-x^7 + x^6 + 20*x^4 - 3*x^3 + 23*x^2 + 4*x + 4)/((x - 1)^4*(x + 1)^4). (End)

cp's OEIS Frontend

A071069 a(n) = min { A070923(k) | n^3 < k < (n+1)^3 }.

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A077109 Duplicate of A070923.

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

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A077110 Nearest integer cube to n^2.

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A181138 Least positive integer k such that n^2 + k is a cube.

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A077107 Least integer cube >= n^2.

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

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A077111 a(n) = A077110(n) - n^2.

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A121536 Smallest m such that m^3 >= n^2.