Special properties of the number $146$

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

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 $N=146$.

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 $146^3$.
• $\underbrace{\color{blue}{11^2 + 44^2 + 66^2}}_{\color{red}{3\text{ terms}}} = \color{blue}{641}\color{red}{3}$