

A259826


Numbers n such that n is a multiple of 6 and both n1 and n+1 are composite.


2



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
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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 "smallgap ktuple" to be an admissible ktuple with all of its gaps in {2,4,6,8}. Every gap G = a(n+1)  a(n) >= 18 contains a smallgap ktuple with k >= G/6  1 and diameter G14, G12 or G10. For example, at n=40 the gap between 1080 and 1134 contains the 9tuple 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(x1)&&!isprime(x+1), vector(10^3, j, 6*j) ) \\ Joerg Arndt, Jul 06 2015
(MAGMA) [n: n in [6..2000 by 6]  not IsPrime(n1) and not IsPrime(n+1)]; // Vincenzo Librandi, Jul 08 2015


CROSSREFS

Intersection of A008588 and A099047.  Michel Marcus, Jul 06 2015
Cf. A060461.
Sequence in context: A074302 A095631 A248662 * A192277 A034291 A039602
Adjacent sequences: A259823 A259824 A259825 * A259827 A259828 A259829


KEYWORD

nonn


AUTHOR

Antonio Gimenez, Jul 05 2015


STATUS

approved



