OFFSET
1,1
COMMENTS
From Brian Almond, Jun 23 2020: (Start)
For every prime gap g, there is a run of consecutive a(n) of length max{[(g+2)/6]-1,0}.
Gaps between successive a(n) correspond to clusters of primes all within +- 8 of each other. The number of primes within a gap G = a(n+1) - a(n) ranges from (G/6 - 1) to (G/6 - 1) plus the number of twin primes within the gap.
Record gaps in a(n) are 24 at a(1)=120, 42 at a(2)=144, 72 at a(10)=342 and 84 at a(1003)=14706 (the next gaps of 84 occur at a(43136164)=369008652 and a(643519601)=5244999552). No larger record gaps exist below 10^10 (n <= 1239026836).
(End)
Define a "small-gap k-tuple" to be an admissible k-tuple with all of its gaps in {2,4,6,8}. Every gap G = a(n+1) - a(n) >= 18 contains a small-gap k-tuple with k >= G/6 - 1 and diameter G-14, G-12 or G-10. For example, at n=40 the gap between 1080 and 1134 contains the 9-tuple p+{0,4,6,10,16,22,30,36,42} for p=1087. - Brian Almond, Jul 25 2020
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..2500
FORMULA
a(n) = 6 * A060461(n). - Brian Almond, Jun 22 2020
EXAMPLE
For n=120, 120 is a multiple of 6, and both 119 and 121 are composite.
MATHEMATICA
Select[6*Range[500], AllTrue[# + {1, -1}, CompositeQ] &] (* Harvey P. Dale, May 21 2017 *)
PROG
(PARI) select(x->!isprime(x-1)&&!isprime(x+1), vector(10^3, j, 6*j) ) \\ Joerg Arndt, Jul 06 2015
(Magma) [n: n in [6..2000 by 6] | not IsPrime(n-1) and not IsPrime(n+1)]; // Vincenzo Librandi, Jul 08 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Antonio Gimenez, Jul 05 2015
STATUS
approved