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

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. 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).

Examples

			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
		

Links

• 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

Crossrefs

Cf. A338776, A234345.

Programs

SageMath

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)])