1.Introduction

Problem of sum of consecutive cubes equal to a cube.
y^3 = (x+1)^3 + (x+2)^3 +....+ (x+n)^3.
C. Pagliani[1] showed the 1000 solutions of this problem.
Many solutions are shown with n<100000. http://oeis.org/A240970

2. Proposition

We consider below diophantine equation.

y^3 = nx^3+3/2n(n+1)x^2+1/2n(n+1)(2n+1)x+1/4n^2(n+1)^2...............................(1)


Equation (1) gives infinitely many integer solutions if n=8m^3 with m is not divisible by 3. 

Let n=8m^3 and y=(2mx+p) then we obtain equation (2).

(96m^6+12m^3-12m^2p)x^2+(4m^3+96m^6+512m^9-6mp^2)x+256m^9+16m^6+1024m^12-p^3=0.......(2)

Since discriminant for x must be perfect square number, then we obtain

V^2 = -12p^4+(384m^4+48m)p^3+(-6144m^8-1152m^5-48m^2)p^2+(49152m^12+768m^6+12288m^9)p
      -131072m^16-49152m^13-5120m^10+16m^4...........................................(3)

We know equation (3) has a solution (p,V) = ( 8m^4+4m^3, 4m^2(2m+1)(2m-1)(8m^4-4m^2-1) ).

Hence we obtain (x,y)=( 1/3(2m+1)(4m^3-8m^2+2m-1), 2/3m(2m-1)(2m+1)(2m^2+1) ).

2m(2m-1)(2m+1)(2m^2+1) is always divisible by 3.

If m=0 mod 3 then (2m+1)(4m^3-8m^2+2m-1)=2 mod 3.
If m=1 mod 3 then (2m+1)(4m^3-8m^2+2m-1)=0 mod 3.
If m=2 mod 3 then (2m+1)(4m^3-8m^2+2m-1)=0 mod 3.

Thus (x,y) is the integer solution if m is not divisible by 3.

n<10000
[  n       x         y ]

[ 64         5      180]
[512       405     5544]
[1000     1133    16830]
[2744     4965    90090]
[4096     8789   175440]
[8000    22533   534660]

C. Pagliani showed  6^3 + 7^3 + ....+ 69^3 = 180^3, 1134^3 + 1135^3 +...+ 2133^3 = 16830^3.




3.Numerical solutions

Search results of the integer points for equation (1) by brute force.
n<10000
Curious solution: [288      272     2856],[343      212     2856]
273^3+274^3+...+560^3=2856^3
213^3+214^3+...+555^3=2856^3

[  n        x       y ]

[  3        2        6]
[  4       10       20]
[ 20        2       40]
[ 25        5       60]
[ 49      290     1155]
[ 64        5      180]
[ 99       10      330]
[125       33      540]
[153      212     1581]
[288      272     2856]
[343      212     2856]
[512      405     5544]
[1000     1133   16830]
[1331     1734   27060]
[1849    34227  431548]
[2197     3605   62244]
[2744     4965   90090]
[4096     8789  175440]
[4913    11367  237456]
[6591      304   82680]
[6859    18170  413820]
[8000    22533  534660]


4.Reference

[1]: L.E. Dickson, History of the Theory of Numbers, Vol.II.