A335225 Partitions of a stronger form of Goldbach Conjecture: the number of decompositions of an even number 2n into the sum of two odd prime numbers p and q with 0 <= q-p < n.

0, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 2, 4, 2, 2, 3, 1, 2, 3, 2, 2, 2, 1, 3, 4, 1, 2, 5, 1, 2, 3, 2, 3, 4, 3, 2, 4, 3, 3, 5, 2, 2, 6, 2, 2, 5, 1, 3, 4, 3, 2, 4, 4, 4, 6, 4, 3, 7, 2, 4, 6, 2, 4, 5, 2, 3, 5, 4, 3, 5, 3, 3, 6, 2, 4, 6, 2, 4, 5, 2, 3, 6, 3

Offset: 1

Author

Ya-Ping Lu, May 27 2020

Keywords

• nonn

Comments

The Goldbach conjecture states that any even number 2n >= 6 can be written as the sum of two unordered odd prime numbers p and q, or 2n = p + q, where 0 <= q-p < 2n.

It appears that the Goldbach conjecture still holds if the span of q-p allowed is reduced by half, from [0, 2n) to [0, n). This stronger form of the Goldbach conjecture is true if a(n) >= 1 for n >= 3. Any further reduction of the q-p span from [0, n) to [0, m), with m < n, results in the number of prime decompositions for at least one of the even numbers being zero.

The values of a(n) and the Goldbach partitions G(n) for n up to 100000 are given in the LINKS section.

Note that a(n) listed above is for the decomposition of even numbers 2n >=6 into unordered odd primes. The sequence for the decomposition of even number 2n >=4 into unordered primes is the same as a(n), except that the second term of the sequence becomes 1.

Examples

			a(1)=0 because 2*1 cannot be written as the sum of two primes.
a(2)=0. Although 2*2 can be written as 2+2, 2 is not an odd prime.
a(3)=1 because 2*3 = 3+3.
a(4)=1 because 2*4 = 3+5.
a(5)=2 because 2*5 = 5+5 and 3+7.
a(6)=1 because 2*6 = 5+7.
a(7)=1 because 2*7 = 7+7. 3+11 is not a valid partition as 11-3 > 7.
		

Links

• Ya-Ping Lu, Values of a(n) and the Goldbach partitions G(n) for n up to 100000

Crossrefs

Cf. A002375, A045917.

Programs

PARI

a(n) = {my(nb=0, m=2*n, q); forprime(p=3, m, if (isprime(q=m-p) && (q%2) && ((q-p)>=0) && ((q-p)Michel Marcus, May 28 2020