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See if there are k-values that work in between.
what is the next cube expressible as a sum of consecutive cubes such that k is a power?
k is a prime number?
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Fun With Num3ers 
If k = x^3 (x not divisible by 3), then m = (x^4 – 3x^3 – 2x^2 + 4) / 6.
The following k = 6591, 21456, 176824, 11859210 are also members of this sequence, yielding the triplets (k, m, cubic root of sum)= (6591, 305, 82680), (21456, 266785, 7715220), (176824, 407526, 28127850), (11859210, 21709458, 6398174475).