Editing Formula for primes (section)

==A function that represents all primes==
Given the constant <math>f_1 = 2.920050977316\ldots</math> {{OEIS|A249270}}, for <math>n \ge 2</math>, define the sequence
{{NumBlk|:|<math>
f_n = \left\lfloor f_{n-1} \right\rfloor
(f_{n-1} - \left\lfloor f_{n-1} \right\rfloor + 1 )
</math>|{{EquationRef|1}}}}
where <math>\left\lfloor\ \right\rfloor</math> is the [[floor function]].
Then for <math>n \ge 1</math>, <math>\left\lfloor f_{n} \right\rfloor</math> equals the <math>n</math>th prime:
<math>\left\lfloor f_1 \right\rfloor = 2</math>,
<math>\left\lfloor f_2 \right\rfloor = 3</math>,
<math>\left\lfloor f_3 \right\rfloor = 5</math>, etc.
<ref name="FridmanEtAl">{{citation
  | doi = 10.1080/00029890.2019.1530554
  | first1=Dylan | last1=Fridman | first2=Juli | last2=Garbulsky | first3=Bruno | last3=Glecer
  | first4=James | last4=Grime | first5=Massi | last5=Tron Florentin
  | title = A Prime-Representing Constant
  | journal = American Mathematical Monthly
  | volume = 126
  | issue = 1
  | pages = 70–73
  | publisher = [[Mathematical Association of America]]
  | location = Washington, DC
  | year = 2019| arxiv=2010.15882 | s2cid=127727922 }}
</ref>
The initial constant <math>f_1 = 2.920050977316</math> given in the article is precise enough for equation ({{EquationNote|1}}) to generate the primes through 37, the <math>12</math>th prime.

The ''exact'' value of <math>f_1</math> that generates ''all'' primes is given by the rapidly-converging [[Series (mathematics)|series]]
:<math>
f_1 = \sum_{n=1}^\infty \frac{p_n - 1}{P_n}
= \frac{2 - 1}{1} + \frac{3 - 1}{2} + \frac{5 - 1}{2 \cdot 3} + \frac{7 - 1}{2 \cdot 3 \cdot 5} + \cdots,
</math>
where <math>p_n</math> is the <math>n</math>th prime, and <math>P_n</math> is the product of all primes less than <math>p_n</math>. The more digits of <math>f_1</math> that we know, the more primes equation ({{EquationNote|1}}) will generate. For example, we can use 25 terms in the series, using the 25 primes less than 100, to calculate the following more precise approximation:
: <math>f_1 \simeq 2.920050977316134712092562917112019.</math>
This has enough digits for equation ({{EquationNote|1}}) to yield again the 25 primes less than 100.

As with Mills' formula and Wright's formula above, in order to generate a longer list of primes, we need to start by knowing more digits of the initial constant, <math>f_1</math>, which in this case requires a longer list of primes in its calculation.