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A335225
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Partitions of a stronger form of Goldbach Conjecture: the number of decompositions of an even number 2n into the sum of two odd prime numbers p and q with 0 <= q-p < n.
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2
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0, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 2, 4, 2, 2, 3, 1, 2, 3, 2, 2, 2, 1, 3, 4, 1, 2, 5, 1, 2, 3, 2, 3, 4, 3, 2, 4, 3, 3, 5, 2, 2, 6, 2, 2, 5, 1, 3, 4, 3, 2, 4, 4, 4, 6, 4, 3, 7, 2, 4, 6, 2, 4, 5, 2, 3, 5, 4, 3, 5, 3, 3, 6, 2, 4, 6, 2, 4, 5, 2, 3, 6, 3
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OFFSET
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1,5
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COMMENTS
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The Goldbach conjecture states that any even number 2n >= 6 can be written as the sum of two unordered odd prime numbers p and q, or 2n = p + q, where 0 <= q-p < 2n.
It appears that the Goldbach conjecture still holds if the span of q-p allowed is reduced by half, from [0, 2n) to [0, n). This stronger form of the Goldbach conjecture is true if a(n) >= 1 for n >= 3. Any further reduction of the q-p span from [0, n) to [0, m), with m < n, results in the number of prime decompositions for at least one of the even numbers being zero.
The values of a(n) and the Goldbach partitions G(n) for n up to 100000 are given in the LINKS section.
Note that a(n) listed above is for the decomposition of even numbers 2n >=6 into unordered odd primes. The sequence for the decomposition of even number 2n >=4 into unordered primes is the same as a(n), except that the second term of the sequence becomes 1.
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LINKS
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EXAMPLE
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a(1)=0 because 2*1 cannot be written as the sum of two primes.
a(2)=0. Although 2*2 can be written as 2+2, 2 is not an odd prime.
a(3)=1 because 2*3 = 3+3.
a(4)=1 because 2*4 = 3+5.
a(5)=2 because 2*5 = 5+5 and 3+7.
a(6)=1 because 2*6 = 5+7.
a(7)=1 because 2*7 = 7+7. 3+11 is not a valid partition as 11-3 > 7.
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PROG
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(PARI) a(n) = {my(nb=0, m=2*n, q); forprime(p=3, m, if (isprime(q=m-p) && (q%2) && ((q-p)>=0) && ((q-p)<n), nb++); ); nb; } \\ Michel Marcus, May 28 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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