<s>[INST] Special properties of the number $146$ I'm a math teacher. Next week I'll give a special lecture about number theory curiosities. It will treat special properties of numbers — the famous story with Ramanujan, taxicab numbers, later numbers divisible by all their digits, etc.
I was given class number $146$ for the lecture and I think it would be fine to start with a special property of our class's number. Ramanujan would surely find something at once, but I can't. Do you see any special properties of $146$?
Here are some of my observations, but these properties are not very special:


*

*$146$ is a semiprime number (product of two distinct primes), while the reversal $641$ is prime.

*$146 = 4^3 + 4^3 + 3^2 + 3^2$.
Here is a very similar question, just to show what kind of question this is and what kind of answers I would like to see. 
 [/INST] The  sum of the first $n$ cubes, starting from $1^3$, is given by the square of the $n$th Triangular number, $T_n$:
\begin{align}
1^3                   &=      & 1^2        &= & T_{1}^2\\
1^3+2^3              &=     & 3^2       &= & T_{2}^2 \\
1^3+2^3+3^3      &=     & 6^2     &= & T_{3}^2 \\
1^3+2^3+3^3+4^3  &= & 10^2   &= & T_{4}^2 \\
\end{align}
Noting this we can write the sum of the first $n$ squares of the Triangular numbers as:
\begin{align*}
\sum_{i=1}^{n}T_i^2&=T_1^2+T_2^2+\dotsb+T_n^2\tag{1}\\
&=n\times1^3+(n-1)\times2^3+\dotsb+1\times n^3\\
&=\sum_{k=1}^{n}(n-k+1)k^3\tag{2}
\end{align*}
Note, taking $n=4$ in $(1)$ gives
$$T_{1}^2+T_{2}^2+T_{3}^2+T_{4}^2=1^2+3^2+6^2+10^2=146$$
and taking $n=4$ in $(2)$ gives
$$4\times1^3+3\times2^3+2\times 3^3+1\times 4^3=146$$
As a consequence of $(1)$ another nice pattern emerges:
$$\frac{1}{4}\left[2^2(1^2+3^2)+4^2(3^2+5^2)\right]=146$$
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