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.

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A265203 Palindromes that can be written as the sum of two or more consecutive positive cubes.

Original entry on oeis.org

9, 99, 9009, 14841, 76167, 108801, 239932, 828828, 886688, 2112112, 4663664, 7152517, 17333371, 17511571, 42844824, 61200216, 135666531, 658808856, 6953443596, 6961551696, 27110501172, 46277277264, 405162261504, 483867768384, 522733337225, 588114411885
Offset: 1

Views

Author

Ann Marie Murray, Dec 03 2015

Keywords

Comments

Can any term in the sequence be written as sum of 2 or more consecutive cubes in more than one way? The answer is no for a(1)-a(46). - Chai Wah Wu, Dec 17 2015

Examples

			14841 can be written as 16^3 + 17^3 + 18^3.

Links

Crossrefs

Cf. A002113, A217843.

Programs

  • Maple
    ispali:= proc(n) local L; L:= convert(n,base,10);
  ListTools:-Reverse(L) = L end proc:
A265203:= proc(N) # get all terms <= N
  local S,a,b,t;
  S:= select(t -> t<=N and ispali(t),
     {seq(seq(b^2*(b+1)^2/4 - a^2*(a+1)^2/4, a=0..b-2),b=2..(1+iroot(4*N,3))/2)});
  sort(convert(S,list));
end proc:
A265203(10^9); # Robert Israel, Dec 07 2015
  • Mathematica
    lim = 800; Sort@ Select[Plus @@@ Map[#^3 &, Select[Flatten[Table[Partition[Range@ lim, k, 1], {k, 2, lim}], 1], Times @@ Differences@ # == 1 &]], # == Reverse@ # &@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 16 2015 *)
  • Sage
    def palindromic_cubic_sums(x_max):
    success = set()
    for x_min in range(1,x_max^(1/3)):
        sum_powers = x_min^3
        for i in range(x_min+1,x_max^(1/3)):
            sum_powers += (i^3)
            if sum_powers >= x_max:
                break
            if str(sum_powers) == str(sum_powers)[::-1]:
                success.add(sum_powers)
    return sorted(success)

A273877 Least positive integer k such that k^3 + (k+1)^3 + ... + (k+n-2)^3 + (k+n-1)^3 is the sum of two positive cubes. a(n) = 0 if no solution exists.

Original entry on oeis.org

0, 1, 11, 2, 10, 31, 6, 70, 4, 42, 4, 4, 15, 174, 6, 2, 70, 556, 18, 378, 2, 119, 4277, 6, 8, 5, 33111, 3, 2088, 61, 7, 7, 145, 417, 8, 13, 9, 1424, 23, 18, 106, 101, 7, 39, 138, 276, 13353, 48, 1, 31, 645, 2981, 107627, 34, 155, 11, 8, 214, 62, 25, 103, 28
Offset: 1

Views

Author

Altug Alkan, Jun 02 2016

Keywords

Comments

What is the most repeated value of this sequence?

Examples

			a(3) = 11 because 11^3 + 12^3 + 13^3 = 7^3 + 17^3.

Links

Crossrefs

Cf. A003325, A217843.

Extensions

a(10)-a(62) from Giovanni Resta, Jun 03 2016
a(49) corrected by Chai Wah Wu, Jun 07 2016

A303383 Total volume of all cubes with side length q such that n = p + q and p <= q.

Original entry on oeis.org

0, 1, 8, 35, 91, 216, 405, 748, 1196, 1925, 2800, 4131, 5643, 7840, 10241, 13616, 17200, 22113, 27216, 34075, 41075, 50336, 59653, 71820, 83916, 99541, 114920, 134603, 153811, 178200, 201825, 231616, 260288, 296225, 330616, 373491, 414315, 464968, 512981
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 22 2018

Keywords

Links

Crossrefs

Cf. A105636.
Subsequence of A217843.
After 8, all terms belong to A265377.

Programs

  • Magma
    [0] cat [&+[(n-k)^3: k in [1..n div 2]]: n in [2..80]]; // Vincenzo Librandi, Apr 23 2018
  • Mathematica
    Table[Sum[(n - i)^3, {i, Floor[n/2]}], {n, 50}]

Formula

a(n) = Sum_{i=1..floor(n/2)} (n-i)^3.
From Bruno Berselli, Apr 23 2018: (Start)
G.f.: x*(1 + x + x^2)*(1 + 6*x + 16*x^2 + 6*x^3 + x^4)/((1 - x)^5*(1 + x)^4).
a(n) = (30*(n - 2)*(n + 1)*(n^2 - n + 2) + (2*n - 1)*(2*n^2 - 2*n - 1)*(-1)^n + 119)/128. Therefore:
a(n) = n^2*(3*n - 2)*(5*n - 6)/64 for n even;
a(n) = (n - 1)^2*(3*n - 1)*(5*n + 1)/64 for n odd. (End)
a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9). - Wesley Ivan Hurt, Apr 23 2021

A338447 Sums of consecutive odd positive cubes.

Original entry on oeis.org

1, 27, 28, 125, 152, 153, 343, 468, 495, 496, 729, 1072, 1197, 1224, 1225, 1331, 2060, 2197, 2403, 2528, 2555, 2556, 3375, 3528, 4257, 4600, 4725, 4752, 4753, 4913, 5572, 6859, 6903, 7632, 7975, 8100, 8127, 8128, 8288, 9261, 10485, 11772, 11816, 12167, 12545, 12888, 13013
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2020

Keywords

Examples

			495 is in the sequence because 495 = 3^3 + 5^3 + 7^3.

Links

Crossrefs

Cf. A002593, A016755, A042965, A217843, A265845, A290276, A320728.

Programs

  • PARI
    lista(nn) = {my(list = List()); forstep (i=1, nn, 2, my(s = 0); forstep(j=i, 1, -2, s += j^3; if (s > nn^3, break); listput(list, s););); Set(list);} \\ Michel Marcus, Nov 13 2020

A343409 Numbers whose square is the sum of one or more consecutive nonnegative cubes.

Original entry on oeis.org

0, 1, 3, 6, 8, 10, 15, 21, 27, 28, 36, 45, 55, 64, 66, 78, 91, 105, 120, 125, 136, 153, 171, 190, 204, 210, 216, 231, 253, 276, 300, 312, 315, 323, 325, 343, 351, 378, 406, 435, 465, 496, 504, 512, 528, 561, 588, 595, 630, 666, 703, 720, 729, 741, 780, 820
Offset: 1

Views

Author

Lamine Ngom, Apr 14 2021

Keywords

Comments

Roots of square terms of A217843. Sequence contains (but is not limited to) cubes (A000578) and triangular numbers (A000217).

Examples

			8 is a term because 8^2 = 64 = 4^3.
10 is a term because 10^2 = 100 = 1^3 + 2^3 + 3^3 + 4^3.

Crossrefs

Cf. A000217, A000578, A126200, A163393, A217843.

Programs

  • Maple
    N:= 1000: # for terms <= N
M:= floor(N^(2/3)):
S:= [seq(n^2*(n+1)^2/4, n=0..M)]:
SD:= {0,seq(seq(S[i]-S[j],j=1..i-1),i=1..M+1)}:
Q:= select(t -> t <= N^2 and issqr(t),SD):
sort(convert(map(sqrt,Q),list)); # Robert Israel, Sep 11 2023

Formula

Union of A000217 and A126200.
Previous Showing 21-25 of 25 results.

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