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.

A048762 Largest cube <= n.

Original entry on oeis.org

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

Views

Author

Charles T. Le (charlestle(AT)yahoo.com)

Keywords

References

  • Krassimir T. Atanassov, On the 40th and 41st Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4, No. 3 (1998), 101-104.
  • J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), 202-204.

Links

Crossrefs

Cf. A048763, A201053, A000578.

Programs

  • Haskell
    a048762 n = last $ takeWhile (<= n) a000578_list
-- Reinhard Zumkeller, Nov 28 2011
  • Maple
    A048762 := proc(n)
        floor(root[3](n)) ;
        %^3 ;
end proc: # R. J. Mathar, Nov 06 2011
  • Mathematica
    Floor[Surd[Range[0,70],3]]^3 (* Harvey P. Dale, Jun 23 2013 *)

Formula

Sum_{n>=1} 1/a(n)^2 = Pi^4/30 + Pi^6/945 + 3*zeta(5). - Amiram Eldar, Aug 15 2022

A201053 Nearest cube.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Nov 28 2011

Keywords

Comments

a(n) = if n-A048763(n) < A048762(n)-n then A048762(n) else A048763(n);
apart from 0, k^3 occurs 3*n^2+1 times, cf. A056107.

Links

Crossrefs

Cf. A061023, A074989, A053187 (nearest square), A000578.
Cf. A048762, A048763, A056107.

Programs

  • Haskell
    a201053 n = a201053_list !! n
a201053_list = 0 : concatMap (\x -> replicate (a056107 x) (x ^ 3)) [1..]
  • Maple
    seq(k^3 $ (3*k^2+1), k=0..10); # Robert Israel, Jan 03 2017
  • Mathematica
    Module[{nn=70,c},c=Range[0,Ceiling[Surd[nn,3]]]^3;Flatten[Array[ Nearest[ c,#]&,nn,0]]] (* Harvey P. Dale, May 27 2014 *)
  • Python
    from sympy import integer_nthroot
def A201053(n):
    a = integer_nthroot(n,3)[0]
    return a**3 if 2*n < a**3+(a+1)**3 else (a+1)**3 # Chai Wah Wu, Mar 31 2021

Formula

G.f.: (1-x)^(-1)*Sum_{k>=0} (3*k^2+3*k+1)*x^((k+1)*(k^2+k/2+1)). - Robert Israel, Jan 03 2017
Sum_{n>=1} 1/a(n)^2 = Pi^4/30 + Pi^6/945. - Amiram Eldar, Aug 15 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

A163849 Primes p such that the difference between the nearest cubes above and below p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 733, 739, 743, 751, 757, 761, 769, 773
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

Keywords

Comments

There is a sequence A048763(A000040(n)) = A145446(n) of nearest cubes above the primes and a sequence A048762(A000040(n)) of nearest cubes below the primes.
If the difference A145446(n) - A048762(A000040(n)) is prime, then A000040(n) is in this sequence.

Examples

			The difference of cubes 6^3 - 5^3 = 91 = 7*13 is not prime, so the primes larger than 5^3 = 125 but smaller than 6^3 = 216 are not in the sequence.

Links

Crossrefs

Cf. A111252, A145446.

Programs

  • Mathematica
    f[n_]:=IntegerPart[n^(1/3)]; lst={};Do[p=Prime[n];If[PrimeQ[(f[p]+1)^3-f[p]^3], AppendTo[lst,p]],{n,6!}];lst

Extensions

Edited by R. J. Mathar, Aug 12 2009

A333884 Difference between smallest cube > n and n.

Original entry on oeis.org

1, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2020

Keywords

Comments

a(n) is the smallest positive number k such that n + k is a cube.

Links

Crossrefs

Cf. A000578, A048763, A055400, A068527, A080883, A154840.

Programs

  • Mathematica
    Table[Floor[n^(1/3) + 1]^3 - n, {n, 0, 80}]

Formula

a(n) = floor(n^(1/3) + 1)^3 - n.

A163850 Primes p such that their distance to the nearest cube above p and also their distance to the nearest cube below p are prime.

Original entry on oeis.org

3, 127, 24391, 29789, 328511, 2460373, 3048623, 9393929, 10503461
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

Keywords

Comments

The two sequences A048763(p) and A048762(p), p=A000040(n), define
nearest cubes above and below each prime p. If p is in A146318, the
distance to the larger cube, A048763(p)-p, is prime. If p is
in the set {3, 11, 13, 19, 29, 67,...,107, 127, 223,..}, the distance to the lower
cube is prime. If both of these distances are prime, we insert p into the sequence.

Examples

			p=3 is in the sequence because the distance p-1=2 to the cube 1^3 below 3, and also the distance 8-p=5 to the cube 8=2^3 above p are prime.
p=127 is in the sequence because the distance p-125=2 to the cube 125=5^3 below p, and also the distance 216-p=89 to the cube 216=6^3 above p, are prime.

Crossrefs

Cf. A163848

Programs

  • Mathematica
    Clear[f,lst,p,n]; f[n_]:=IntegerPart[n^(1/3)]; lst={};Do[p=Prime[n];If[PrimeQ[p-f[p]^3]&&PrimeQ[(f[p]+1)^3-p],AppendTo[lst,p]],{n,9!}];lst
dncQ[n_]:=Module[{c=Floor[Surd[n,3]]},AllTrue[{n-c^3,(c+1)^3-n},PrimeQ]]; Select[Prime[Range[230000]],dncQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 16 2016 *)

Extensions

Edited, first 5 entries checked by R. J. Mathar, Aug 12 2009
Two more terms (a(8) and a(9)) from Harvey P. Dale, Oct 16 2016
Showing 1-6 of 6 results.

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