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

3, 53, 157, 173, 211, 257, 263, 373, 509, 541, 563, 593, 607, 653, 733, 947, 977, 997, 1069, 1103, 1123, 1187, 1223, 1237, 1367, 1459, 1499, 1511, 1543, 1747, 1753, 1759, 1777, 1901, 1907, 1913, 2069, 2179, 2287, 2399, 2411, 2417, 2447, 2677, 2903, 2963, 3061, 3067, 3181, 3203, 3307, 3313, 3511

Offset: 1

Author

Marc Morgenegg, Sep 18 2017

Keywords

• nonn

Comments

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.

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

Examples

			3 is a term, because 3+2 = 5 and 3+5 = 8; neither 5 nor 8 is divisible by 3.
53 is a term as well, because 53+47 = 100 and 53+59 = 112, and neither 100 nor 112 is divisible by 3.
		

Programs

Maple

Primes:= select(isprime,[2,seq(i,i=3..10000,2)]):
R:= select(k -> Primes[k]+Primes[k-1] mod 3 <> 0, {$2..nops(Primes)}):
R:= R intersect map(`-`,R,1);
Primes[sort(convert(R,list))]; # Robert Israel, Sep 18 2017

Mathematica

Select[Prime@ Range@ 500, NoneTrue[# + {NextPrime[#, -1], NextPrime@ #}, Divisible[#, 3] &] &] (* Michael De Vlieger, Sep 19 2017 *)

PARI

isok(p) = isprime(p) && ((p + precprime(p-1)) % 3) && ((p + nextprime(p+1)) % 3) \\ Michel Marcus, Sep 18 2017

Extensions

More terms from Robert Israel, Sep 18 2017