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.

Solution 1:

$146$ can be written as squares of two primes: $$146=5^2+11^2=5^2+(1+4+6)^2=(1+4)^2+(1+4+6)^2.$$

Solution 2:

The Postmaster General has decided that only three different stamp denominations shall be produced and also that one may stick at most ten stamps on an envelope. Also, it shall be possible to stick stamps of any total value from $1$ cent, $2$ cents, ..., up to $N$ cents (inclusive). Of course, the value of $N$ depends on the three stamp denominations. What is $N$ if the denominations are $1$ cent, $10$ cents, $15$ cents? Who can find a better choice of stamp denominations? What is the best choice of stamp denominations and what is $N$ for that choice?

The answer to the last question id $N=146$.

Solution 3:

Interesting question... Here are a few facts about $146$ I just found...

• $146 = (1^3 - 1) - 4^3 + (6^3 - 6)$
• $(1^2 + 4^2 + 6^2) + (1+4+6) = 4^3$.
• $641 - 146$ is divisible by $1+4+6$
• The sum of the sum of digits of $146^1, 146^4, 146^6$ is $2^7-1$.
• The sum of the product of the digits of $146^2$ and $146^3$ is $12^2$, which is $146 - 2$.
• The digits, with repetition, of $146^2$ are all contained in the digits of $146^3$.
• $\underbrace{\color{blue}{11^2 + 44^2 + 66^2}}_{\color{red}{3\text{ terms}}} = \color{blue}{641}\color{red}{3}$

Solution 4:

Two nice facts about sums of cubes:

1. We can write $146$ as $$4 \cdot 1^3 + 3 \cdot 2^3 +2 \cdot 3^3 + 1 \cdot 4^3.$$ The previous term in this sequence is $$3 \cdot 1^3 + 2 \cdot 2^3 + 1 \cdot 3^3 = 46.$$
2. $146$ is the second-largest integer that can't be written as the sum of cubes bigger than $1$. (The largest is $154$.)

Also, http://oeis.org/A134907 gives us a really weird formula for $146$: $$146 = \left\lfloor 5 e^{-\tan 5}\right\rfloor.$$

From http://oeis.org/A172877, we see that there are exactly $146$ $4 \times 2$ matrices such as $$\begin{bmatrix}1 & 4 \\ 2 & 3 \\ 3 & 2 \\ 4 & 1\end{bmatrix}$$ with nonnegative integer entries in which each row sums to $5$ and each column sums to $10$.

Solution 5:

Other than $1$, $10$ and $100$ which may be considered trivial cases, $146$ is the smallest positive integer $n$ such that, ignoring sequence and repetitions, $n^2$ and $n^3$ contain exactly the same digits:

$$146^2 = 21316$$

$$146^3 = 3112136$$

This is readily verified by reviewing the values of $n^2$ and $n^3$ for $n = 2,3,...,145$.

Addendum 11 June 2018

In fact this is the only non-trivial case of a positive integer with this property for $n < 1000$.