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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 18 11:30 EDT 2021. Contains 343995 sequences. (Running on oeis4.)