factorizations of $n!$ for $1 < n < 50$
FactorizationsOfNFor1N50
2008-04-22 21:11:29
2008-04-22 21:11:29
Example
de Polignac's formula
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The following table gives the exponent for each of the primes from 2 to 47 in the prime factorization of $n!$. Stripping off the zeroes and ignoring the top row and the leftmost column, this table is A115627 in Sloane's OEIS.
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}
$n$& 2 & 3 & 5 & 7 & 11& 13& 17& 19& 23& 29& 31& 37& 41& 43&47 \\
2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
3 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
5 & 3 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
6 & 4 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
7 & 4 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 7 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
9 & 7 & 4 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
10 & 8 & 4 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
11 & 8 & 4 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
12 & 10 & 5 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
13 & 10 & 5 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
14 & 11 & 5 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
15 & 11 & 6 & 3 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
16 & 15 & 6 & 3 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
17 & 15 & 6 & 3 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
18 & 16 & 8 & 3 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
19 & 16 & 8 & 3 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
20 & 18 & 8 & 4 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
21 & 18 & 9 & 4 & 3 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
22 & 19 & 9 & 4 & 3 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
23 & 19 & 9 & 4 & 3 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
24 & 22 & 10 & 4 & 3 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
25 & 22 & 10 & 6 & 3 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
26 & 23 & 10 & 6 & 3 & 2 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
27 & 23 & 13 & 6 & 3 & 2 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
28 & 25 & 13 & 6 & 4 & 2 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
29 & 25 & 13 & 6 & 4 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
30 & 26 & 14 & 7 & 4 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
31 & 26 & 14 & 7 & 4 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
32 & 31 & 14 & 7 & 4 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
33 & 31 & 15 & 7 & 4 & 3 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
34 & 32 & 15 & 7 & 4 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
35 & 32 & 15 & 8 & 5 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
36 & 34 & 17 & 8 & 5 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
37 & 34 & 17 & 8 & 5 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
38 & 35 & 17 & 8 & 5 & 3 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
39 & 35 & 18 & 8 & 5 & 3 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
40 & 38 & 18 & 9 & 5 & 3 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
41 & 38 & 18 & 9 & 5 & 3 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
42 & 39 & 19 & 9 & 6 & 3 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
43 & 39 & 19 & 9 & 6 & 3 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
44 & 41 & 19 & 9 & 6 & 4 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
45 & 41 & 21 & 10 & 6 & 4 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
46 & 42 & 21 & 10 & 6 & 4 & 3 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 0 \\
47 & 42 & 21 & 10 & 6 & 4 & 3 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\
48 & 46 & 22 & 10 & 6 & 4 & 3 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\
49 & 46 & 22 & 10 & 8 & 4 & 3 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{tabular}