
COMMENTS

This sequence gives numbers n such that k^3 + (k+1)^3 + ... + (k+n1)^3 = y^3 for some nontrivial k.  Derek Orr, Jan 18 2015
For n > 3, n appears not to be squarefree. [This is incorrect, 25201 = 11*29*79 is a member of this sequence.  Derek Orr, Mar 09 2015]
5000 < a(17) <= 6591.  Derek Orr, Jan 17 2015
It is known that 14161 and 63001 are members of this sequence.
Let S(n,k) = sum{i=0..n1}((k+i)^3). The smallest kvalues for the given n in the sequence are {1, 3, 11, 3, 6, 291, 6, 11, 34, 213, 273, 213, 406, 1134, 1735, 34228, 3606, 4966, 8790, 11368}.
If n is of the form (2m)^3 for some m = 1, 2, 3, ... there exists only a finite number of k that can make S(n,k) be a cube. For k large enough, S(n,k)  floor(S(n,k)^(1/3))^3 = m^3*(n^21)(2k + n  1). So it will never be a cube.
If n is of the form (2m+1)^3 for some m = 1, 2, 3, ... there also exists only a finite number of k that can make S(n,k) cube. For k large enough, S(n,k)  floor(S(n,k)^(1/3))^3 = f(m)*n(2k + n  1) where f(m) is a function of m. f(1) = 91, f(2) = 1953, f(3) = 14706, f(4) = 66430, f(5) = 221445, f(6) = 603351, ...
As mentioned above, when k is greater than a certain number, S(n,k) will never be a cube. Let the critical value for k be called g(n), since it depends on n. The function g(n) is only concretely known when n is a cube (mentioned above "for k large enough"). Below are the known g(n) values.
n = 8, g(n) <= 6.
n = 27, g(n) <= 168.
n = 64, g(n) <= 1333.
n = 125, g(n) <= 6447.
n = 216, g(n) <= 23219.
n = 343, g(n) <= 68456.
n = 512, g(n) <= 174506.
n = 729, g(n) <= 398215.
n = 1000, g(n) <= 832832.
n = 1331, g(n) <= 1623264.
n = 1728, g(n) <= 2985119.
n = 2197, g(n) <= 5227943.
...
Example: for n = 8, if k > 6, then S(8,k)  floor(S(8,k)^(1/3))^3 = 63*(2*k+7). So g(n) = 6.
There is a possibility that g(n) could be defined for other values of n besides the cubes and this would give a bound on k for all n. Note that g(n) ~ 0.065*n^(2.4). Graphing this function, one can see that for all n > 4 listed in the sequence, their kvalues given in the first comment are well less than 0.065*n^(2.4). For large n, 0.065*n^(2.4) seems to be a large overestimate. Thus, 0.065*n^(2.4) can act as a sufficient bound for k for large n (essentially for n > 4).
If n = v^3 where v is a number not divisible by 3, then k = (v^43*v^32*v^2+4)/6. Thus there are infinitely many cubes in this sequence.  Robert Israel, Aug 06 2014
The equations S(n,k) = y^3 is a superelliptic curve and thus has only a finite number of integral points.
Using the transformation X = (2*y)/(2k  1 + n) and Y = 1/(2k  1 + n), S(n,k) = y^3 transforms to Y^2*(n^3  n) = X^3  n, which is in Weierstrass form and can be solved for rationals (X,Y).
Further, the above transformation can be manipulated to arrive at a new diophantine equation; this one is over the integers: n*X^3 + (n^3  n)*X = (2y)^3 where X = 2k + n  1.
The Bilu and Hanrot link gives explicit bounds for the possible values of x for the equation a*y^p = f(x) for p >= 3 and f of degree d >= 2. Here, a = 1, p = 3, and d = 3.
If M is a cubed integer not divisible by 3, there always exists at least one nontrivial solution for the sum of M consecutive cubed integers starting from a^3 and equaling a cubed integer c^3. There are no nontrivial solutions for M = m^3 if m=0(mod 3). For n>=1, for integers m(n) = A001651(n), all nontrivial solutions for M(n) = m^3 = A118719(n+1) are a(n) = (m1)(m^2(m2)  4(m+1))/6 (A253778) and c(n)= m(m^21)(m^2+2)/6 (A253779).  Vladimir Pletser, Jan 12 2015
The above relation for the number of terms equal to a cube not 0(mod 3), i.e., a(n) = (m1)(m^2(m2)  4(m+1))/6 (V. Pletser) or k = (v^43*v^32*v^2+4)/6 (R. Israel), was already given by A. Martin in 1871 and J. Matteson in 1888 (see L. E. Dickson, p. 584).  Vladimir Pletser, Jan 27 2015
Additional terms in this sequence (that are not cubes) are n = 6591, 21456, 176824, 11859210, yielding the triplets (n, first term, cubic root of sum) = (6591, 305, 82680), (21456, 266785, 7715220), (176824, 407526, 28127850), (11859210, 21709458, 6398174475).  Vladimir Pletser, Jan 12 2015
Two additional terms in this sequence (that are not cubes) are n = 13923 and 33124 yielding the triplets (n, first term, cubic root of sum) = (13923, 3010, 273819) and (33124, 18551, 1205750) (found in K. S. Brown's Mathpages).  Vladimir Pletser, Feb 01 2015
Another solution (when n is not a cube) is (n, first term, cubic root of sum) = (25201, 46690, 1764070). Note that n = 25201 = 11*29*79 is squarefree. This is the first known squarefree solution after n = 3 (given in the Math Overflow link).  Derek Orr, Mar 09 2015
A118719 (excluding its initial term) is a subsequence.  Derek Orr, May 12 2015
