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

1,5

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

Links

Lei Zhou, Table of n, a(n) for n = 1..10000

Lei Zhou, Plot for 0<n<=200000.

Index entries for sequences related to Goldbach conjecture

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A023201, A045917, A199920, A219055.

Sequence in context: A025819 A243866 A110102 * A230198 A024939 A024937

Adjacent sequences: A255469 A255470 A255471 * A255473 A255474 A255475

Keyword

nonn,easy

Author

Lei Zhou, Feb 23 2015

Status

approved