A259826 Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.

120, 144, 186, 204, 216, 246, 288, 300, 324, 342, 414, 426, 474, 516, 528, 534, 552, 582, 624, 636, 666, 696, 714, 780, 792, 804, 816, 834, 846, 870, 894, 900, 924, 960, 1002, 1026, 1044, 1056, 1074, 1080, 1134, 1140, 1146, 1158, 1176, 1206, 1242, 1254, 1266, 1272, 1314, 1332, 1338, 1344, 1350

Offset: 1

Author

Antonio Gimenez, Jul 05 2015

Keywords

• nonn

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

Examples

			For n=120, 120 is a multiple of 6, and both 119 and 121 are composite.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..2500

Crossrefs

Intersection of A008588 and A099047. - Michel Marcus, Jul 06 2015

Cf. A060461.

Programs

Magma

[n: n in [6..2000 by 6] | not IsPrime(n-1) and not IsPrime(n+1)]; // Vincenzo Librandi, Jul 08 2015

Mathematica

Select[6*Range[500], AllTrue[# + {1, -1}, CompositeQ] &] (* Harvey P. Dale, May 21 2017 *)

PARI

select(x->!isprime(x-1)&&!isprime(x+1), vector(10^3,j,6*j) ) \\ Joerg Arndt, Jul 06 2015

Formula

a(n) = 6 * A060461(n). - Brian Almond, Jun 22 2020