You might look at OEIS A271116. It also generates all primes and some pseudoprimes to base 3. I don't know how the timing /efficiency compares to your method, but you can see the Mathematica code there. Ref: https://oeis.org/A271116

There is a more interesting form:

E^((4 I Pi (1 + 2^x x)) / (-3 + (-1)^x - 2 x))

which doesn't require floor's and gives you1s for primes:

{1, 1, 1, E^((2*I*Pi)/3), 1, 1, E^((2*I*Pi)/5), 1, 1, E^((2*I*Pi)/7), 1, E^(-((4*I*Pi)/5)), E^((8*I*Pi)/9), 1, 1,  E^((2*I*Pi)/11), E^((16*I*Pi)/35), 1, E^((2*I*Pi)/13), 1, 1, E^(-((2*I*Pi)/3)), 1, E^((4*I*Pi)/7), E^((2*I*Pi)/17), 1, E^(-((14*I*Pi)/55)), E^((2*I*Pi)/19), 1, 1, E^((2*I*Pi)/21), E^((6*I*Pi)/13), 1, E^((2*I*Pi)/23), 1, 1, E^((22*I*Pi)/25), E^((16*I*Pi)/77), 1, E^((26*I*Pi)/27), 1, E^((6*I*Pi)/17), E^((2*I*Pi)/29), 1, E^(-((8*I*Pi)/13)), E^((2*I*Pi)/31), E^(-((84*I*Pi)/95)), 1, E^(-((28*I*Pi)/33)), 1}

This was found by replacing floor with the equivalent Fourier derived square-wave, aligned on the integer line.

Oh, this I did not notice. OK, my complete code for your function now reads:

pQ[0] = pQ[1] = 0; (* not prime *)
pQ[2] = 1;  (* prime *)
 pQ[k_] := If[OddQ[k], 
  With[{n = k - 1}, -Floor[(2 (-1 + 2^n n))/(3 - (-1)^n + 2 n)] + Floor[(2 (1 + 2^n n))/(3 - (-1)^n + 2 n)]], 0]

Looks great! But if you try more numbers, I am afraid it does not work, e.g.:

Select[{#, pQ[#] == Boole[PrimeQ[#]]} & /@ Range[1000], (! Last[#]) &]
(*  Out:   {{341,False},{561,False},{645,False}}   *)

Chris,

in the link to "linkedin.com" you gave you are talking about "primality testing" - this is a test for prime numbers, right? But using your formula I cannot see any correlation with prime numbers:

pQ[n_] := -Floor[(2 (-1 + 2^n n))/(3 - (-1)^n + 2 n)] + Floor[(2 (1 + 2^n n))/(3 - (-1)^n + 2 n)]

Where am I wrong?