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%I #29 Sep 08 2022 08:46:13
%S 120,144,186,204,216,246,288,300,324,342,414,426,474,516,528,534,552,
%T 582,624,636,666,696,714,780,792,804,816,834,846,870,894,900,924,960,
%U 1002,1026,1044,1056,1074,1080,1134,1140,1146,1158,1176,1206,1242,1254,1266,1272,1314,1332,1338,1344,1350
%N Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.
%C From _Brian Almond_, Jun 23 2020: (Start)
%C For every prime gap g, there is a run of consecutive a(n) of length max{[(g+2)/6]-1,0}.
%C 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.
%C 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).
%C (End)
%C 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
%H Harvey P. Dale, <a href="/A259826/b259826.txt">Table of n, a(n) for n = 1..2500</a>
%F a(n) = 6 * A060461(n). - _Brian Almond_, Jun 22 2020
%e For n=120, 120 is a multiple of 6, and both 119 and 121 are composite.
%t Select[6*Range[500], AllTrue[# + {1, -1}, CompositeQ] &] (* _Harvey P. Dale_, May 21 2017 *)
%o (PARI) select(x->!isprime(x-1)&&!isprime(x+1), vector(10^3,j,6*j) ) \\ _Joerg Arndt_, Jul 06 2015
%o (Magma) [n: n in [6..2000 by 6] | not IsPrime(n-1) and not IsPrime(n+1)]; // _Vincenzo Librandi_, Jul 08 2015
%Y Intersection of A008588 and A099047. - _Michel Marcus_, Jul 06 2015
%Y Cf. A060461.
%K nonn
%O 1,1
%A _Antonio Gimenez_, Jul 05 2015
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