## Combinatorics, factorial, primes and the L-sequence!

Combinatorics is, amongst others, about counting discrete structures. Factorial, used in Newton’s symbol, is big part of it. It is also directly connected to the primes – it is only about multiplication, and multiplication is used for counting elements (iteration of addition). And L-sequence is the next step to revise it. Every integer $k$ can be represented as $k = p_1^{a_1}p_2^{a_2}..p_n^{a_n}$. L-sequence is $[a_1,..,a_n]$ for factorials. Below the first values. If you think about it, factorial is also the closest possible thing to Riemann’s Zeta function! Connecting the Zeta function with the analysis of behavior of factorials might be crucial to understanding the distribution of primes.

From there on, I am creating the factorial triangle.

1. Find the formula that describe row values in the triangle.
2. Notice that in the column associated with $i$-th prime, we have $i$ times 1, then $i$ times 2,.., would that be true in general?

Now, look at places when “new” primes come into play:

And finally, at sequences – how are those sequences related?

1,2,3,4,5,6,7,.. (missing some prime)

1,1,2,2,3,3,4,4,5,5,…(missing some prime,prime)

1,1,1,2,2,2,3,3,3,4,4,4,…(missing some prime,prime,prime)

(1,1,..,1)(2,2..,2),..(missing some prime,…,prime)