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A259826
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Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.
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2
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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
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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For n=120, 120 is a multiple of 6, and both 119 and 121 are composite.
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MATHEMATICA
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Select[6*Range[500], AllTrue[# + {1, -1}, CompositeQ] &] (* Harvey P. Dale, May 21 2017 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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