In "The Riemann Hypothesis", Notices of the AMS, March 2003, J. B. Conrey points out that
**g(k) = (k^2)! * prod (j=0, k-1, j!/(j+k)!)** has an interesting prime factorisation.

I have discovered (or re-discovered) that if **v(g(k), p(n))** denotes the power to which
the nth prime **p(n)** is raised in the prime factorisation of **g(k)** then it appears that the following partial symmetry holds

```
v(g(k), p(n)) = v(g(p(n) - k), p(n)), n > 1, 1 <= k <= (p(n) – 1)/2
```

To see the full extent of the partial symmetry, each prime factorisation needs to be infinite in extent.

The partial symmetry can clearly be seen in the portion of the table of exponents below defined by **1 <= k < p(n)**, **2 < p(n) <= 23**.

```
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
p(n) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79
k
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 3 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 2 1 1 0 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
6 4 2 1 0 2 2 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0
7 4 1 1 1 1 2 2 2 2 1 1 1 1 1 1 0 0 0 0 0 0 0
8 7 2 2 1 0 1 3 3 2 2 2 1 1 1 1 1 1 1 0 0 0 0
9 6 4 3 1 0 1 3 4 3 2 2 2 1 1 1 1 1 1 1 1 1 1
10 7 4 4 3 0 0 2 4 4 3 3 2 2 2 2 1 1 1 1 1 1 1
11 6 5 4 3 1 0 2 3 5 4 3 3 2 2 2 2 1 1 1 1 1 1
12 10 7 6 2 1 0 1 2 5 4 4 3 3 3 3 2 2 2 2 2 1 1
13 9 9 6 3 1 1 0 1 4 5 5 4 4 3 3 3 2 2 2 2 2 2
14 11 9 5 4 1 1 0 1 3 6 6 5 4 4 4 3 3 3 2 2 2 2
15 11 8 5 4 2 1 0 0 2 6 7 6 5 5 4 4 3 3 3 3 3 2
16 15 8 5 5 4 1 0 0 2 5 7 6 6 5 5 4 4 4 3 3 3 3
17 14 7 4 6 4 2 1 0 1 4 6 7 7 6 6 5 4 4 4 4 3 3
18 15 8 3 7 3 2 1 0 1 4 5 8 7 7 6 6 5 5 4 4 4 4
19 14 7 3 8 2 4 1 1 0 3 4 8 8 8 7 6 6 5 5 5 4 4
20 17 6 4 9 3 4 1 1 0 2 3 7 9 9 8 7 6 6 5 5 5 5
21 15 8 3 10 3 3 1 1 0 2 3 6 9 10 9 8 7 7 6 6 6 5
22 17 7 3 10 4 3 2 1 0 1 2 6 8 10 10 9 8 7 7 6 6 6
```

**v(g(k), p(n))** can be quickly calculated by noting that

```
v(g(k), p(n)) = (sum(j=0, k-1; sod(j+k, p(n)) - sod(j, p(n))) - sod(k^2, p(n)))/(p(n) - 1)
```

where **sod(m, p(n))** is the sum of the digits of **m** in base **p(n)**.

I have not seen this conjecture mentioned in the literature to which I have access so I would be most grateful if someone could point me in the right direction to learn more about it.