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

6, 6, 6, 6, 12, 18, 18, 30

Offset: 2

Author

Cino Hilliard, Apr 18 2004

Keywords

• nonn
• more

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 n-th 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.

Examples

			For n = 3, we have the difference between nextprime and precprime for 2k <= 10^3:
   2k | occurrences
  -----------------
    2 |  35
    4 |  80
    6 | 132
    8 |  60
   10 |  80
   12 |  44
   14 |  49
   16 |   0
   18 |   9
   20 |  10
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.
		

Crossrefs

Cf. A060267.

Programs

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*n-1); \\ A060267
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

Extensions

Edited by Michel Marcus, Sep 16 2020