2,1

The density of primes occurring with these numbers A060267(2n) 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 k-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 2n <= 10^10 will be 30 or higher. This appears as a plausable statement since as 2n increases, the probability that the difference between the next and preceding prime will contain larger and larger prime factors.

For k = 3 we have the difference between nextprime and precprime for 2n<=10^3:

2n 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 2n <= 10^3. Thus 6 has the maximum

number of occurrences and is the second entry in the table.

(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) }

Cf. A060267.

Sequence in context: A082509 A103337 A001734 this_sequence A116571 A054641 A024731

Adjacent sequences: A092934 A092935 A092936 this_sequence A092938 A092939 A092940

uned,nonn

Cino Hilliard (hillcino368(AT)gmail.com), Apr 18 2004