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 1-7 of 7 results.

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

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A000290, A077110, A077119.

Formula

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

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

Views

Author

Zak Seidov and Reinhard Zumkeller, Oct 15 2002

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

Extensions

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

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

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

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Examples

			a(5)=121, as 121=11^2 is the nearest square to 125=5^3.

Crossrefs

Cf. A000290, A000578, A002821, A077119, A077110.

Programs

  • Mathematica
    Table[Round[Sqrt[n^3]]^2, {n, 0, 39}] (* Alonso del Arte, Dec 07 2011, based on Artur Jasinski's program for A077119 *)
  • Python
    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

Formula

a(n) = if A077116(n) < A070929(n) then A065733(n) else A077115(n).
a(n) = A002821(n)^2. - Chai Wah Wu, Jul 30 2022

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

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

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

A077112 a(n)=n^2 times nearest cube to n^2.

Original entry on oeis.org

0, 1, 4, 72, 128, 675, 972, 3136, 4096, 5184, 12500, 15125, 18000, 21125, 42336, 48600, 55296, 99127, 111132, 123823, 137200, 225792, 247808, 270848, 294912, 455625, 492804, 531441, 571536, 613089, 900000, 961000, 1024000, 1089000
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 29 2002

Keywords

Links

Programs

  • Mathematica
    ntnc[n_]:=Module[{n2=n^2,a,b,a3,b3},a=Floor[Power[n2,(3)^-1]]; b=a+1; a3=a^3; b3=b^3;If[n2-a3Harvey P. Dale, Jan 06 2012 *)
nic[n_]:=Module[{n2=n^2,c},c=Floor[Surd[n2,3]];n2*Nearest[{c^3,(c+1)^3}, n2]]; Join[{0,1},Array[nic,40,2]]//Flatten//Union (* Harvey P. Dale, May 13 2020 *)

Formula

a(n) = A000290(n)*A077110(n).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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