
COMMENTS

The density of primes occurring with these numbers A060267(2k) appears to max out at higher and higher values of 6x. For example, looking at numbers in the sequence for next and prec prime differences <= 50, we have the following table for nth powers of 10.
k max density
2 6  21
3 6  132
4 6  897
5 6  5820
6 12  48030
7 18  394659
8 18  3462648
9 30  32669865
Conjecture: The maximum density occurs at increasing multiples of 6 as the number of primes tested approaches infinity. E.g. the number of nextprime  precprime occurrences for 2k <= 10^10 will be 30 or higher. This appears as a plausable statement since as 2k increases, the probability that the difference between the next and preceding prime will contain larger and larger prime factors.


PROG

(PARI) prmppr(n) = { mx=0; f = vector(floor(sqrt(n)+2)); forstep(x=4, n, 2, y=nextprime(x)precprime(x); print1(y", "); if(y>mx, mx=y); f[y]++; ); print(); mx2=0; forstep(x=2, mx, 2, if(f[x] > mx2, mx2=f[x]; d=x); print(x", "f[x]); ); print(d", "mx2) } \\ use prmppr(1000) to get a(3)=6
(PARI) f(n) = nextprime(2*n+1)  precprime(2*n1); \\ A060267
a(n) = {my(v=vector(10^n/21, k, f(k+1))); my(nbm = 0, imax = 0); forstep (i=vecmin(v), vecmax(v), 2, my(nb = #select(x>(x==i), v)); if (nb > nbm, nbm = nb; imax = i); ); imax; } \\ Michel Marcus, Sep 16 2020
