A338776 a(n) = card(GB(2*n)), where GB(n) is the set of primes which are Goldbach-associated with n.

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

Offset: 0

Author

Peter Luschny, Nov 08 2020

Keywords

• nonn

Comments

For an integer n >= 0 we say a prime p is gb-associated with n if sqrt(n) < p <= n/2 and no prime q which is <= sqrt(n) divides p*(p - n). Let GB(n) be the set of integers which are gb-associated with n (for examples see A338777). a(n) is the number of primes which are gb-associated with n.

If a(n) > 0 for n >= 3 then Goldbach's conjecture is true.

Examples

			Comparison of the sets whose cardinality is given by A002375(n) resp. a(n).
m  A002375          A338776
32 [29, 19]         [19]
34 [31, 29, 23, 17] [23, 17]
36 [31, 29, 23, 19] [29, 23, 19]
38 [31, 19]         [31, 19]
		

Links

• Peter Luschny, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A338777, A002375, A244408.

Programs

SageMath

# [using gb_associated from A338777]
def A338776(n):
    return len(gb_associated(2*n))
print([A338776(n) for n in range(87)])

Formula

a(n) <= A002375(n).

a(n) = A002375(n) <=> n in A244408 (for n >= 2).