A338776 -id:A338776 - cp's OEIS frontend

A338777 a(n) = Product_{k in GB(2*n)} k, where GB(n) is the set of primes which are Goldbach-associated with n.

1, 1, 1, 3, 3, 5, 5, 7, 5, 35, 7, 55, 385, 91, 11, 1001, 13, 187, 1547, 133, 187, 2717, 91, 391, 24871, 247, 253, 55913, 247, 5423, 2800733, 589, 4301, 164749, 31, 124729, 2442583, 14911, 11339, 4075291, 9139, 300817, 2629420651, 10621, 20213, 116883421171, 7657

Offset: 0

Peter Luschny, Nov 08 2020

nonn

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. Then a(n) = Product_{k in GB(2*n)} k.

If a(n) != 1 for n >= 3 then Goldbach's conjecture is true. In this case m = max(GB(2*n)) exists and P = (2*n - m, m) is a Goldbach partition of 2*n (cf. A234345).

			m:  GB(m)  -> Product(GB)
0:   []    -> 1
2:   []    -> 1
4:   []    -> 1
6:   [3]   -> 3
8:   [3]   -> 3
10:  [5]   -> 5
...
90:  [11, 17, 19, 23, 29, 31, 37, 43] -> 116883421171
92:  [13, 19, 31]                     -> 7657
94:  [11, 23, 41, 47]                 -> 487531
96:  [13, 17, 23, 29, 37, 43]         -> 234524537
98:  [19, 31, 37]                     -> 21793
100: [11, 17, 29, 41, 47]             -> 10450121

Peter Luschny, Table of n, a(n) for n = 0..1000
Denise Vella-Chemla, Continuer de suivre Galois, 2013.
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A338776, A234345.

def gb_associated(n):
    r = isqrt(n)
    A = prime_range(2, r + 1)
    B = prime_range(r + 1, n // 2 + 1)
    return [p for p in B if all((p * (p - n) % q) != 0 for q in A)]
def A338777(n):
    return prod(gb_associated(2*n))
print([A338777(n) for n in range(47)])

cp's OEIS Frontend

A338777 a(n) = Product_{k in GB(2*n)} k, where GB(n) is the set of primes which are Goldbach-associated with n.

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