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A338777
a(n) = Product_{k in GB(2*n)} k, where GB(n) is the set of primes which are Goldbach-associated with n.
2
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
0,4
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).
EXAMPLE
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
PROG
(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)])
CROSSREFS
Sequence in context: A002374 A261046 A226482 * A110560 A172170 A233808
KEYWORD
nonn
AUTHOR
Peter Luschny, Nov 08 2020
STATUS
approved