#### Primorials:

While reading up on the unlikelyhood of odd perfect numbers, I ran across the primorial function. Combine *prime* and *factorial*: **primorial**! That is:

For N primorial, we write N#. To calculate N#, we take $J$ such that $p_k \leq$ N.

Thus, 10# = 9# = 8# = 7# $= 2 \times 3 \times 5 \times 7$.

And of course, Wikipedia's article on primorials (O beloved reference!).

#### Odd Perfect Numbers:

Summing up Pomerance's Heuristic that Odd Perfect Numbers are Unlikely (link above): We can calculate the probability that a large perfect number ($n: n = pm^2 > 10^{300}$) ($10^{300}$ chosen because no odd perfects have been found in the interval $[1,10^{300}]$) exists. For a large, arbitrary $m$, hence, $m^2$, the probability is really small. It so happens that for even perfect numbers we can get a much better estimate on $m$ than simply arbitrary (thus, increasing the probability we can find large even perfect numbers, which do exist. But, at the same time, we really have not gotten a good handle on $m$ for odd primes (see the other threads on Perfect numbers).

Primorials come in to play in getting an upper bound for $\sigma(m^2)$ (for $N$ to be perfect, $p| \sigma(m^2)$). The upper bound is then used as an estimate in the nasty probability integral done in the calculations.

#### Bonus Material I Came Across:

$p_n$# +/- 1 is sometimes a prime.

$p_n$# $\pm 1$ are twin primes for $n = 3, 5$ (the OEIS lists were too short to determine if there were any more overlaps (for $n > 1829$, there are none $5 < n < 1829$)).

Highly composite numbers (in the sense that the HCN is the smallest integer such that $\nu(HCN)$ increases to a record) are products of primorials.

Many HCNs are formed by multiplying together smaller HCNs. New primes are introduced on some HCNs, such as 2 (2), 6 (3), 60 (5), and 840 (7). For some HCNs, a prime is "re-introduced", such as 1260 (7*180).