<s>[INST] numbers' pattern It is known that 
$$\begin{array}{ccc}1+2&=&3 \\ 4+5+6 &=& 7+8 \\
9+10+11+12 &=& 13+14+15 \\\
16+17+18+19+20 &=& 21+22+23+24 \\\
25+26+27+28+29+30 &=& 31+32+33+34+35 \\\ldots&=&\ldots
\end{array}$$
There is something similar for square numbers:
$$\begin{array}{ccc}3^2+4^2&=&5^2 \\ 10^2+11^2+12^2 &=& 13^2+14^2 \\ 21^2+22^2+23^2+24^2 &=& 25^2+26^2+27^2 \\ \ldots&=&\ldots
\end{array}$$
As such, I wonder if there are similar 'consecutive numbers' for cubic or higher powers.
Of course, we know that there is impossible for the following holds (by Fermat's last theorem):
$$k^3+(k+1)^3=(k+2)^3 $$
 [/INST] We do have
$$ \eqalign{6^3 &= 3^3 + 4^3 + 5^3\cr
            20^3 &= 11^3 + 12^3 + 13^3 + 14^3\cr
            40^3 &= 3^3 + \ldots + 22^3\cr
            70^3 &= 15^3 + \ldots + 34^3\cr
37^3 + 38^3 &= 5^3 + \ldots + 25^3\cr
  30^3 + 31^3 + 32^3 &= 7^3 + \ldots + 24^3\cr 
            101^3 + 102^3 + 103^3 &= 61^3 + \ldots + 71^3\cr
 15^3 + \ldots + 20^3 &= 11^3 + \ldots + 19^3\cr
            681^3 + \ldots + 687^3 &= 566^3 + \ldots + 577^3\cr
 \cr}$$
and many others.
EDIT: See also http://oeis.org/A062682
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