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Numbers n such that n!!-8 and n!!+8 are two consecutive primes.
0

%I #4 Jun 25 2013 16:50:05

%S 41,91,299,341

%N Numbers n such that n!!-8 and n!!+8 are two consecutive primes.

%C n is in the sequence iff n!!-8 and n!!+8 are primes and the four numbers n!!-2,n!!+2, n!!-4 & n!!+4 are composites. It's interesting that related to the term 341 we can say 341!!-(3+4+1) and 341!!+(3+4+1) are two consecutive primes with length 341+prime(3+4+1). Conjecture: 341 is the largest term of this sequence.

%e 341 is in the sequence because 341!!-8 and 341!!+8 are primes and the four numbers 341!!-2, 341!!+2, 341!!-4 & 341!!+4 are composites.

%t Select[Range[5,500],PrimeQ[ #!!-8]&&PrimeQ[ #!!+8]&&!PrimeQ[ #!!-4] &&!PrimeQ[ #!!+4]&&!PrimeQ[ #!!-2]&&!PrimeQ[ #!!+2]&]

%t n8pQ[n_]:=Module[{x=n!!},PrimeQ[x-8]&&NextPrime[x-8]==x+8]; Select[Range[ 350],n8pQ] (* _Harvey P. Dale_, Jun 25 2013 *)

%Y Cf. A076188.

%K nonn

%O 1,1

%A _Farideh Firoozbakht_, Nov 19 2006