Are there any integers which could never be a Godel Number?

Question

In a system where a number such as $121$ can be encoded into a Godel number, are there any integers which could never be the result?

As I understand it, we line up the primes and raise these to the digit we wish to encode and then multiply these together to find the Godel encodeing of number.

So $121$ is encoded as follows:

$2^1 * 3^2 * 5^1 \implies 2 * 9 *5 = 90$

So $90$ is the Godel number for $121$.

I am interested to know what integers (if any) could never be a Godel number. Further more, I would like to know if this set of numbers has a name and if they have anything else in common or are useful for anything.

There are many Gödel numbering(s): in general, the encoding rule has the property that from the code we can retrieve the formula but not every number will be a code. — Mauro ALLEGRANZA, Feb 11 '22 at 14:39
See the post What does a Godel sentence actually look like? and this post for some "exercises" in encoding. — Mauro ALLEGRANZA, Feb 11 '22 at 14:41
Because the first digit cannot be zero, Godel numbers are always even, never odd. — Geoffrey Trang, Feb 11 '22 at 15:08
There are Gödel numberings such that any natural number is a Gödel number. — Wuestenfux, Feb 11 '22 at 16:00

score 0 · Accepted Answer · answered Feb 11 '22 at 15:18

Any odd integer (unless leading zeros are allowed) and any integer for which there is at least one prime dividing it ten or more times could never be a Godel number.

In particular, $2^{10}=1024$ cannot be a Godel number, because only digits $0$ to $9$ are allowed.

In case of a negative integer $n$, one should assume that the Godel number of $n$ is just $-m$, where $m$ is the Godel number of $-n$, so that the question still makes sense for negative integers.

Are there any integers which could never be a Godel Number?

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