TITLE: Divisors of sequence $1!+2!+\ldots+n!$
QUESTION [7 upvotes]: Is the set of prime numbers which divide at least one number in the sequence $a_n=1!+2!+\ldots+n!$ finite or infinite?
I try to show that it's infinite. Suppose the set is finite and consists of $p_1<p_2<\ldots<p_k$. Consider $n>p_k$. We have $a_n=1!+2!+\ldots+p_k!+(p_k+1)!\ldots+n!$. The terms from $(p_k+1)!$ onward are divisible by $p_1,p_2,\ldots,p_k$. The problem is that we do not have control over the power of $p_i$ that divides $1!+2!+\ldots+p_k!$ and $(p_k+1)!\ldots+n!$.

REPLY [2 votes]: This sequence gets a mention in Guy's Unsolved Problems in Number Theory (3-rd edition, 2004); denoted $K_n$ below:

So I suspect you might not get a conclusive answer.  I would guess it's infinite.
Here's a table for small values:
$$\begin{array}{|r|r|}
\hline
n & \text{prime factorization of } a_n \\
\hline
1 &  1  \\
2 &  3  \\
3 &  3 \cdot 3  \\
4 &  3 \cdot 11  \\
5 &  3 \cdot 3 \cdot 17  \\
6 &  3 \cdot 3 \cdot 97  \\
7 &  3 \cdot 3 \cdot 3 \cdot 3 \cdot 73  \\
8 &  3 \cdot 3 \cdot 11 \cdot 467  \\
9 &  3 \cdot 3 \cdot 131 \cdot 347  \\
10 &  3 \cdot 3 \cdot 11 \cdot 40787  \\
11 &  3 \cdot 3 \cdot 11 \cdot 443987  \\
12 &  3 \cdot 3 \cdot 11 \cdot 11 \cdot 23 \cdot 20879  \\
13 &  3 \cdot 3 \cdot 11 \cdot 821 \cdot 83047  \\
14 &  3 \cdot 3 \cdot 11 \cdot 2789 \cdot 340183  \\
15 &  3 \cdot 3 \cdot 11 \cdot 107 \cdot 509 \cdot 259949  \\
16 &  3 \cdot 3 \cdot 11 \cdot 225498914387  \\
17 &  3 \cdot 3 \cdot 11 \cdot 163 \cdot 20143 \cdot 1162943  \\
18 &  3 \cdot 3 \cdot 11 \cdot 19727 \cdot 3471827581  \\
19 &  3 \cdot 3 \cdot 11 \cdot 29 \cdot 43 \cdot 1621 \cdot 641751001  \\
20 &  3 \cdot 3 \cdot 11 \cdot 11 \cdot 53 \cdot 67 \cdot 662348503367  \\
21 &  3 \cdot 3 \cdot 11 \cdot 877 \cdot 3203 \cdot 41051 \cdot 4699727  \\
22 &  3 \cdot 3 \cdot 11 \cdot 11895484822660898387  \\
23 &  3 \cdot 3 \cdot 11 \cdot 139 \cdot 2129333 \cdot 922459185301  \\
24 &  3 \cdot 3 \cdot 11 \cdot 37 \cdot 37 \cdot 29131483 \cdot 163992440081  \\
25 &  3 \cdot 3 \cdot 11 \cdot 454823 \cdot 519472957 \cdot 690821017  \\
26 &  3 \cdot 3 \cdot 11 \cdot 107 \cdot 173 \cdot 7823 \cdot 12227 \cdot 1281439 \cdot 1867343  \\
27 &  3 \cdot 3 \cdot 11 \cdot 431363 \cdot 2882477797 \cdot 91865833117  \\
28 &  3 \cdot 3 \cdot 11 \cdot 191 \cdot 47793258077 \cdot 349882390108241  \\
29 &  3 \cdot 3 \cdot 11 \cdot 37 \cdot 283 \cdot 5087 \cdot 1736655143086866180331  \\
30 &  3 \cdot 3 \cdot 11 \cdot 2771826449193354891007108898387  \\
\hline
\end{array}$$
One might even suspect that $a_n$ is $99$ times a prime infinitely often (see Sloane's http://oeis.org/A122990).