A332656 Number of decompositions of 2n into unordered sums of two odd primes, including at least one twin prime.

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

Offset: 1

Author

Harry E. Neel, Feb 18 2020

Keywords

• nonn

Comments

a(n) is the number of ways that a twin prime may be summed with another prime to provide the even number 2n. It is not known if summing primes with twin primes will provide every even number greater than 4 (or greater than or equal to 6).

Examples

			a(6) = 1 because the only way to express 2*6 = 12 as the sum of two primes, one of which is a twin prime, is 5+7. (Since the sequence counts unordered sums, 7+5 is not counted as distinct from 5+7.)
Also, 37+61 = 98 is a valid sum, 61 being a part of a twin prime pair; while 37+47 = 84 is not a valid sum because neither 37 nor 47 is a part of a twin prime pair.
		

Crossrefs

Cf. A000040, A001097, A002375, A129363.

Programs

PARI

istwin(p) = isprime(p-2) || isprime(p+2);
a(n) = {n *= 2; my(nb = 0, q, v=[]); forprime(p=2, n, q = n-p; if ((q>=p) && isprime(q) && (istwin(p) || istwin(q)), nb++; v= concat(v, p));); nb;} \\ Michel Marcus, Feb 28 2020