Willans' formula is a prime-generating formula due to Willan (1964) that is defined as follows. Let
(1)
(2)
for
an integer, where
is the floor function. This formula is a consequence
of Wilson's theorem and conceals the prime numbers
as those for which , i.e., the values of are 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, ... (OEIS A080339).
Then
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Rowland,
E. "An Exact Formula for the Primes: Willans' Formula." Sept. 22,
2022. https://www.youtube.com/watch?v=j5s0h42GfvM.Sloane,
N. J. A. Sequence A080339 in "The
On-Line Encyclopedia of Integer Sequences."Willans, C. P.
"A Formula for the th Prime Number." Math. Gaz.48, 413-415,
1964.
![|_cos^2[pi((j-1)!+1)/j]_|](/images/equations/WillansFormula/Inline3.svg)
an integer, where 
is the floor function. This formula is a consequence
of Wilson's theorem and conceals the prime numbers
as those for which 
, i.e., the values of 
are 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, ... (OEIS A080339).
Then
is the prime counting function (Willans
1964; Havil 2003, pp. 168-169).
th Prime Number." Math. Gaz. 48, 413-415,
1964.