A321221 Numbers of the form 6n-2 which are not a sum of two numbers that are the lesser of twin primes.

4, 94, 400, 514, 784, 904, 1114, 1144, 1264, 1354, 3244, 4204

Offset: 1

Author

Marcin Barylski, Oct 31 2018

Keywords

• nonn
• more

Comments

Conjecture: This sequence is finite.

If this sequence is finite, then the Goldbach Strong Conjecture is true. If p1 and p2 are the lesser of twin primes, then q1=p1+2 and q2=p2+2 are also primes (they are the greater of twin primes). If 6n-2 = p1+p2, then 6n = q1+p2 and 6n+2 = q1+q2.

Examples

			a(1) = 4 because 4 = 2+2; there are no other Goldbach partitions and 2 is not the lesser of twin primes.
a(2) is not 6 because 6 = 3+3 and 3 is the lesser of twin primes.
		

Links

• Marcin Barylski, On 6k ± 1 Primes in Goldbach Strong Conjecture
• Marcin Barylski, C++ program for generating A321221

Crossrefs

Cf. A001359, A002375.

Programs

Mathematica

aQ[n_]:= (k=1; kmax=(n+2)/6; While[k<=kmax && !AllTrue[{6k-1,6k+1,6(kmax-k)-1,6(kmax-k)+1}, PrimeQ], k++]; k>kmax); Select[6*Range[0,10000]+4,aQ] (* Amiram Eldar, Nov 10 2018 *)

PARI

ok(n)={if(n%6 == 4, forstep(k=3, n\2, 2, if(isprime(k) && isprime(k+2) && isprime(n-k) && isprime(n-k+2), return(0))); 1, 0)} \\ Andrew Howroyd, Nov 01 2018