A255472 Number of decompositions of 2n into sums of two primes p <= q such that one or both p and q are elements of A023201.

0, 0, 0, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 1, 4, 4, 1, 3, 4, 3, 3, 4, 4, 3, 5, 3, 3, 6, 1, 5, 6, 2, 5, 5, 4, 5, 6, 4, 4, 8, 4, 3, 8, 3, 5, 7, 2, 5, 7, 5, 6, 6, 6, 6, 9, 5, 4, 12, 3, 5, 10, 2, 5, 7, 6, 5, 6, 6, 5, 11, 5, 4, 11, 2, 8, 8, 3, 7, 10, 5, 4, 9, 8, 5

Offset: 1

Author

Lei Zhou, Feb 23 2015

Keywords

• nonn
• easy

Comments

Conjecture: for all n > 3, a(n) > 0.

If 2n = p + q and p+6 is also a prime, 2n+6 can be written as the sum of two primes p+6 and q.

The conjecture is weaker than a conjecture of Sun posed in 2012 (see A219055). - Zhi-Wei Sun, Mar 18 2015

Examples

			n=4: 2n=8=3+5, 5+6=11 is also a prime number. This is the only occurrence, so a(4)=1.
n=5: 2n=10=3+7=5+5. Both 5+6=11 and 7+6=13 are prime numbers. Two occurrences found, so a(5)=2.
		

Links

• Lei Zhou, Table of n, a(n) for n = 1..10000
• Lei Zhou, Plot for 0
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A023201, A045917, A199920, A219055.

Programs

Mathematica

Table[e = 2 n; ct = 0; p1 = 2; While[p1 = NextPrime[p1]; p1 <= n, p2 = e - p1; If[PrimeQ[p2], If[PrimeQ[p1 + 6] || PrimeQ[p2 + 6], ct++]]]; ct, {n, 1, 100}]

PARI

a(n) = {nb = 0; forprime(p=2, 2*n, if ((q=2*n-p) && (q <= p) && isprime(q=2*n-p) && (isprime(q+6) || isprime(p+6)), nb++);); nb;} \\ Michel Marcus, Mar 01 2015