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