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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 (list; graph; refs; listen; history; text; internal format)
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).

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

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

Cf. A338776, A234345.

Sequence in context: A002374 A261046 A226482 * A110560 A172170 A233808

Adjacent sequences:  A338774 A338775 A338776 * A338778 A338779 A338780

KEYWORD

nonn

AUTHOR

Peter Luschny, Nov 08 2020

STATUS

approved

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Last modified April 11 18:00 EDT 2021. Contains 342888 sequences. (Running on oeis4.)