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A092937 Differences nextprime(2k) - precprime(2k) having maximum prime density for 2k <= 10^n.

%I A092937 #18 Sep 19 2020 18:26:17
%S A092937 6,6,6,6,12,18,18,30
%N A092937 Differences nextprime(2k) - precprime(2k) having maximum prime density for 2k <= 10^n.
%C A092937 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 n-th powers of 10.
%C A092937   k| max|  density
%C A092937   2|  6 |       21
%C A092937   3|  6 |      132
%C A092937   4|  6 |      897
%C A092937   5|  6 |     5820
%C A092937   6| 12 |    48030
%C A092937   7| 18 |   394659
%C A092937   8| 18 |  3462648
%C A092937   9| 30 | 32669865
%C A092937 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.
%e A092937 For n = 3, we have the difference between nextprime and precprime for 2k <= 10^3:
%e A092937    2k | occurrences
%e A092937   -----------------
%e A092937     2 |  35
%e A092937     4 |  80
%e A092937     6 | 132
%e A092937     8 |  60
%e A092937    10 |  80
%e A092937    12 |  44
%e A092937    14 |  49
%e A092937    16 |   0
%e A092937    18 |   9
%e A092937    20 |  10
%e A092937 6 occurs 132 times in the differences for 2k <= 10^3. Thus 6 has the maximum number of occurrences and is the second entry in the table. So a(3) = 6.
%o A092937 (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
%o A092937 (PARI) f(n) = nextprime(2*n+1) - precprime(2*n-1); \\ A060267
%o A092937 a(n) = {my(v=vector(10^n/2-1, 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
%Y A092937 Cf. A060267.
%K A092937 nonn,more
%O A092937 2,1
%A A092937 _Cino Hilliard_, Apr 18 2004
%E A092937 Edited by _Michel Marcus_, Sep 16 2020