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%I #14 Nov 09 2020 08:35:06
%S 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,
%T 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,
%U 4,4,6,6,5,9,5,4,11,3,6,9,4,6,8,4,5,11
%N a(n) = card(GB(2*n)), where GB(n) is the set of primes which are Goldbach-associated with n.
%C 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.
%C If a(n) > 0 for n >= 3 then Goldbach's conjecture is true.
%H Peter Luschny, <a href="/A338776/b338776.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) <= A002375(n).
%F a(n) = A002375(n) <=> n in A244408 (for n >= 2).
%e Comparison of the sets whose cardinality is given by A002375(n) resp. a(n).
%e m A002375 A338776
%e 32 [29, 19] [19]
%e 34 [31, 29, 23, 17] [23, 17]
%e 36 [31, 29, 23, 19] [29, 23, 19]
%e 38 [31, 19] [31, 19]
%o (SageMath) # [using gb_associated from A338777]
%o def A338776(n):
%o return len(gb_associated(2*n))
%o print([A338776(n) for n in range(87)])
%Y Cf. A338777, A002375, A244408.
%K nonn
%O 0,10
%A _Peter Luschny_, Nov 08 2020
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