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A272367 Primes p separated from their adjacent primes on both sides by a prime number of successive composites, while the adjacent primes of p are separated by a prime number of integers.

%I A272367 #13 Nov 15 2016 12:15:35
%S A272367 53,89,97,113,127,157,173,211,257,263,307,317,331,359,367,373,389,397,
%T A272367 401,449,457,479,487,491,499,509,541,563,593,607,653,683,727,733,743,
%U A272367 751,761,769,773,853,863,877,887,907,911,937,947,953,967,977,983,991,997,1069,1103,1109,1117,1123,1187
%N A272367 Primes p separated from their adjacent primes on both sides by a prime number of successive composites, while the adjacent primes of p are separated by a prime number of integers.
%H A272367 Charles R Greathouse IV, <a href="/A272367/b272367.txt">Table of n, a(n) for n = 1..10000</a>
%e A272367 a(1) = 53. The primes around and including 53 are {47, 53, 59}. The number of composites between these are {5, 5} and the number of integers between 47 and 59 is 11; all of {5, 5, 11} are prime, thus 53 is a term.
%t A272367 Select[Prime@ Range@ 195, Function[p, Times @@ Boole@ PrimeQ@ Flatten[Map[Differences, {#, Delete[#, 2]}] - 1] &@ Map[NextPrime[p, #] &, Range[-1, 1]] == 1]] (* _Michael De Vlieger_, Apr 27 2016 *)
%o A272367 (PARI) list(lim)=my(v=List(),p=2,q=3); forprime(r=5,nextprime(lim\1+1), if(isprime(q-p-1) && isprime(r-q-1) && isprime(r-p-1), listput(v,q)); p=q; q=r); Vec(v) \\ _Charles R Greathouse IV_, Apr 30 2016
%Y A272367 Subsequence of A209617.
%K A272367 nonn
%O A272367 1,1
%A A272367 _Marc Morgenegg_, Apr 27 2016
%E A272367 More terms from _Michael De Vlieger_, Apr 27 2016