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A092937 Differences nextprime(2n) - precprime(2n) having maximum prime density for 2n <= 10^k. 0
6, 6, 6, 6, 12, 18, 18, 30 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=2..9.

EXAMPLE

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.

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) }

CROSSREFS

Cf. A060267.

Sequence in context: A001734 A173067 A233550 * A285287 A285048 A265830

Adjacent sequences:  A092934 A092935 A092936 * A092938 A092939 A092940

KEYWORD

uned,nonn

AUTHOR

Cino Hilliard, Apr 18 2004

STATUS

approved

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Last modified June 2 09:21 EDT 2020. Contains 334769 sequences. (Running on oeis4.)