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A292530 Primes prime(k) such that neither prime(k) + prime(k-1) nor prime(k) + prime(k+1) is divisible by 3.

%I A292530 #35 Aug 29 2020 13:12:28
%S A292530 3,53,157,173,211,257,263,373,509,541,563,593,607,653,733,947,977,997,
%T A292530 1069,1103,1123,1187,1223,1237,1367,1459,1499,1511,1543,1747,1753,
%U A292530 1759,1777,1901,1907,1913,2069,2179,2287,2399,2411,2417,2447,2677,2903,2963,3061,3067,3181,3203,3307,3313,3511
%N A292530 Primes prime(k) such that neither prime(k) + prime(k-1) nor prime(k) + prime(k+1) is divisible by 3.
%C A292530 Prime(k) is the k-th prime. It seems to be rare that the sum of two consecutive primes is not divisible by 3. For each prime(k) in this sequence (other than prime(2) = 3), the three numbers prime(k-1), prime(k), and prime(k+1) are all of the form 6*x+1 or all of the form 6*x-1.
%C A292530 Apart from the first term a(1) = 3 also middle of 3 consecutive primes whose sum is divisible by 3. - _Hugo Pfoertner_, Aug 29 2020
%e A292530 3 is a term, because 3+2 = 5 and 3+5 = 8; neither 5 nor 8 is divisible by 3.
%e A292530 53 is a term as well, because 53+47 = 100 and 53+59 = 112, and neither 100 nor 112 is divisible by 3.
%p A292530 Primes:= select(isprime,[2,seq(i,i=3..10000,2)]):
%p A292530 R:= select(k -> Primes[k]+Primes[k-1] mod 3 <> 0, {$2..nops(Primes)}):
%p A292530 R:= R intersect map(`-`,R,1);
%p A292530 Primes[sort(convert(R,list))]; # _Robert Israel_, Sep 18 2017
%t A292530 Select[Prime@ Range@ 500, NoneTrue[# + {NextPrime[#, -1], NextPrime@ #}, Divisible[#, 3] &] &] (* _Michael De Vlieger_, Sep 19 2017 *)
%o A292530 (PARI) isok(p) = isprime(p) && ((p + precprime(p-1)) % 3) && ((p + nextprime(p+1)) % 3) \\ _Michel Marcus_, Sep 18 2017
%K A292530 nonn
%O A292530 1,1
%A A292530 _Marc Morgenegg_, Sep 18 2017
%E A292530 More terms from _Robert Israel_, Sep 18 2017