A Paper from 2009 found an interesting connection to the Taylor Series https://browse.arxiv.org/pdf/0709.0671.pdf
I have found a seperate function that generates these primes as well
[ a(n) = n \cdot a(n-1) + 1 ] [ a(n) \mod p = (n \cdot a(n-1) + 1) \mod p ]
- Recursive Sequence: Each term depends on the previous term.
- Multiplicative Structure: Involves multiplication by integers.
- Addition of 1: Each term is incremented by 1.
- Defined for All Natural Numbers: The sequence is generated for all ( n ).
[ S = 1 - 1! + 2! - 3! + \ldots + (-1)^{p-1} \cdot (p-1)! ] [ S \mod p = \sum_{i=0}^{p-1} (-1)^i \cdot i! \mod p ]
Despite being fundamentally different, both ( a(n) ) and ( S ) yield the same modulo ( p ) values for prime ( p ).
..... This formula feels very similar to another interesting set of relationships:
[ u_{n+1} = -1 + (n+1)u_n ] initialized with: [ u_0 = e - 1 ] ... this (when symbolically calculated) forms:
- Integers (( n )): The index ( n ) in the sequence progression represents natural counting, starting from 0.
- The sequence generates terms that feature factorial growth when associated with the number ( e ). Specifically, the coefficient of ( e ) in each term is ( n! ), indicating that the term captures the factorial nature of ( n ), scaled by ( e ).
- The negative component in each term of the sequence signifies the total number of arrangements possible for a set of ( n ) distinct objects. These arrangements include all permutations of subsets of an ( n )-set, which is also the number of one-to-one sequences that can be formed from ( n ) distinct objects.
- The sequence for these negative components is: 1, 2, 5, 16, 65, 326, 1957, 13700, ...
The negative components can be understood as the sum of permutations of all subsets of an ( n )-set, including the empty set. This can be defined mathematically as: [ a(n) = \sum_{k=0}^{n} \frac{n!}{k!} ] This is the total number of one-to-one sequences that can be formed from ( n ) distinct objects, including sequences with zero elements (i.e., the empty set).
sequence encapsulates two distinct mathematical properties:
- Factorial Growth: The positive part of each term captures the factorial nature of ( n ), scaled by ( e ).
- Combinatorial Counting: The negative component of each term represents the total number of permutations of all subsets of an ( n )-set, following the sum ( a(n) = \sum_{k=0}^{n} \frac{n!}{k!} ).
This sequence is a fascinating blend of both exponential and combinatorial growth, making it rich for analysis in both calculus and combinatorial mathematics.