I computed the primes b4+1 in [2; 10,000,000]. Because N = 4, the gaps are small and we can 'see' the distribution.
There are 445868 primes in this range and 35321 gap=2, 36565 gap=4, 37619 gap=6, 26814 gap=8, ..., 1 gap=276.
I compared it with the exponential distribution:
D(b) = pi . (exp(2/lambda) - 1) . exp(-b/lambda)
where pi = 445322.4 (the expected number of primes) and lambda = 10000000 / pi ~ 22.45.
In blue the prime gap frequency distribution, in orange the exponential distribution.