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 1-3 of 3 results.

A097812 Numbers n such that n^2 is the sum of two or more consecutive positive squares.

Original entry on oeis.org

5, 29, 70, 77, 92, 106, 138, 143, 158, 169, 182, 195, 245, 253, 274, 357, 385, 413, 430, 440, 495, 531, 650, 652, 655, 679, 724, 788, 795, 985, 1012, 1022, 1055, 1133, 1281, 1365, 1397, 1518, 1525, 1529, 1546, 1599, 1612, 1786, 1828, 2205, 2222, 2257, 2372
Offset: 1

Views

Author

T. D. Noe, Aug 25 2004

Keywords

Comments

These numbers were found by exhaustive search. The sums are not unique; for n = 143, there are two representations. The Mathematica code prints n, the range of squares in the sum and the number of squares in the sum. Because the search included sums of all squares up to 2000, this sequence is complete up to 2828.

Examples

			29 is in this sequence because 20^2 + 21^2 = 29^2.
Contribution from _Donovan Johnson_, Feb 19 2011: (Start)
For seven terms < (10^15)^(1/2), the square is a sum in two different ways:
143^2 = 7^2 + ... + 39^2 = 38^2 + ... + 48^2.
2849^2 = 294^2 + ... + 367^2 = 854^2 + ... + 864^2.
208395^2 = 2175^2 + ... + 5199^2 = 29447^2 + ... + 29496^2.
2259257^2 = 9401^2 + ... + 25273^2 = 26181^2 + ... + 32158^2.
6555549^2 = 41794^2 + ... + 58667^2 = 87466^2 + ... + 92756^2.
11818136^2 = 10898^2 + ... + 74906^2 = 29929^2 + ... + 76392^2.
19751043^2 = 39301^2 + ... + 107173^2 = 249217^2 + ... + 255345^2. (End)

Links

Crossrefs

Cf. A097811 (n^3 is the sum of consecutive cubes).
Cf. A001032, A151557.

Programs

  • Mathematica
    g[m0_, m1_] := (1 - m0 + m1)(-m0 + 2m0^2 + m1 + 2m0 m1 + 2m1^2)/6; A097812 = {}; Do[n = g[m0, m1]^(1/2); If[IntegerQ[n], Print[{n, m0, m1, m1 - m0 + 1}]; AppendTo[A097812, n]], {m1, 2, 2000}, {m0, m1 - 1, 1, -1}]; Union[A097812]

Extensions

Name edited by Altug Alkan, Dec 07 2015

A131643 Cubes that are also sums of three or more consecutive positive cubes.

Original entry on oeis.org

216, 8000, 64000, 216000, 343000, 5832000, 35937000, 157464000, 1540798875, 3951805941, 22069810125, 23295638016, 58230605376, 170400029184, 4767078987000, 19814511816000, 241152896222784, 565199024832000, 731189187729000, 5399901725184000, 13389040129314816, 15517248640897024
Offset: 1

Views

Author

Tanya Khovanova, Sep 08 2007

Keywords

Comments

Note that by Fermat's theorem no cube is the sum of two positive cubes.
All entries have the form A000537(j) - A000537(i-1) with 1 <= i < j, for example (j,i) = (5,3), (14,11), (22,3), (30,6), (34,15), (69,6), (109,11). - R. J. Mathar, Sep 14 2007 [Presumably this comment refers just to the terms shown, and not to every term in the sequence. - N. J. A. Sloane, Dec 19 2015]
Subsequence of A265845 (numbers that are sums of consecutive positive cubes in more than one way) which is sparse: among the first 1000 terms of A265845, only 17 are cubes. - Jonathan Sondow, Jan 10 2016

Examples

			216 = 27 + 64 + 125.
Note that "positive" is needed in the definition, otherwise the sequence would contain 8 = (-1)^3 + 0^3 + 1^3 + 2^3. - _N. J. A. Sloane_, Dec 19 2015

Links

Crossrefs

a(n) = A097811(n)^3. - Donovan Johnson, Nov 09 2012
Cf. A000537, A265845.

Programs

  • Mathematica
    Select[Union[ Flatten[Table[ Plus @@ Table[i^3, {i, k, j}], {k, 1000}, {j, k + 1, 1000}]]], # <= 1000^3 && IntegerQ[ #^(1/3)] &]

Extensions

More terms from R. J. Mathar, Sep 14 2007
More terms from Donovan Johnson, Mar 09 2008
Name edited by Jon E. Schoenfield, Dec 07 2015

A307609 Number of partitions of n^3 into consecutive positive cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 18 2019

Keywords

Examples

			20^3 = 11^3 + 12^3 + 13^3 + 14^3, so a(20) = 2.
2856^3 = 213^3 +...+ 555^3 = 273^3 +...+ 560^3, so a(2856) = 3. See also _Donovan Johnson_'s comment in A097811. - _Antti Karttunen_, Aug 22 2019

Links

Crossrefs

Cf. A000578, A030272, A097811, A131643, A217843, A259792, A297199.

Programs

  • PARI
    A297199(n) = { my(s=0, k=1, c); while((c=k^3) <= n, my(u=n-c, i=k); while(u>0, i++; c = i^3; u=u-c); s += (!u); k++); (s); };
A307609(n) = A297199(n^3); \\ Antti Karttunen, Aug 22 2019

Formula

a(n) = [x^(n^3)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^3).
a(n) = A297199(A000578(n)).
a(n) >= 2 for n in A097811.
Showing 1-3 of 3 results.

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