cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Benoit Cloitre, May 20 2002

Keywords

Comments

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

Links

Crossrefs

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

Formula

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

Extensions

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

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

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Examples

			a(10) = 125, as 125 = 5^3 is the nearest cube to 100 = 10^2.

Links

Crossrefs

Cf. A000578, A000290, A075847, A070923, A077106, A077107, A077111, A077118.
Cf. A002760 (Squares and cubes). - Zak Seidov, Oct 08 2015

Programs

  • Mathematica
    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 *)
  • Python
    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

Formula

a(n) = if A075847(n) < A070923(n) then A077106(n) else A077107(n).

A077106 Largest integer cube <= n^2.

Original entry on oeis.org

0, 1, 1, 8, 8, 8, 27, 27, 64, 64, 64, 64, 125, 125, 125, 216, 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
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Examples

			a(20) = 343, as 343 = 7^3 is the largest cube <= 400 = 20^2.

Links

Crossrefs

Cf. A000578, A000290, A065733, A075847, A077107, A077110, A077113.

Programs

  • Mathematica
    a[n_] := Floor[Surd[n^2, 3]]^3; Array[a, 100, 0] (* Amiram Eldar, Apr 05 2025 *)

Formula

a(n) + A075847(n) = n^2.
a(n) = A100196(n)^3. - Amiram Eldar, Apr 06 2025

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

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Examples

			a(10) = 1024, as 1024 = 32^2 is the least square >= 1000 = 10^3.

Crossrefs

Cf. A000290, A000578, A048761, A065733, A077107, A077118, A185549.

Programs

  • Magma
    [Ceiling(n^(3/2))^2: n in [0..50]]; // Vincenzo Librandi, Feb 17 2015
  • Mathematica
    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 *)

Formula

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

A185549 a(n) = ceiling(n^(3/2)); complement of A185550.

Original entry on oeis.org

0, 1, 3, 6, 8, 12, 15, 19, 23, 27, 32, 37, 42, 47, 53, 59, 64, 71, 77, 83, 90, 97, 104, 111, 118, 125, 133, 141, 149, 157, 165, 173, 182, 190, 199, 208, 216, 226, 235, 244, 253, 263, 273, 282, 292, 302, 312, 323, 333, 343, 354, 365, 375, 386, 397, 408, 420, 431, 442, 454, 465, 477, 489
Offset: 0

Views

Author

Clark Kimberling, Jan 30 2011

Keywords

Links

Crossrefs

Cf. A000093, A077107, A185550.
Cf. A144968 (first differences).

Programs

  • Haskell
    a185549 = ceiling . (** (3 / 2)) . fromIntegral
-- Reinhard Zumkeller, Jul 24 2015
  • Mathematica
    f[n_]=Ceiling[n^(3/2)];
t1=Table[f[n],{n,1,90}];t1  (* A185549 *)
t2=Complement[Range[150], Table[f[n],{n,1,80}]];t2  (* A185550 *)
Ceiling[Sqrt[Range[0,70]^3]] (* Harvey P. Dale, Jul 04 2023 *)
  • PARI
    for(n=0,50, print1(ceil(n^(3/2)), ", ")) \\ G. C. Greubel, Jul 08 2017

Formula

a(n) = ceiling(n^(3/2)).
a(n) = sqrt(A077107(n)). - Amiram Eldar, May 17 2025

Extensions

Initial a(0)=0 prepended and offset adjusted by Reinhard Zumkeller, Jul 24 2015

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

Views

Author

Zak Seidov, Aug 05 2006

Keywords

Crossrefs

Cf. A070923, A077107, A100196.

Programs

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

Formula

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

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, May 17 2025
Showing 1-6 of 6 results.

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