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A307774
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Number of partitions of n into 3 parts such that the middle and largest parts are both prime.
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0
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0, 0, 0, 0, 1, 2, 2, 2, 3, 2, 4, 2, 3, 3, 5, 4, 6, 3, 5, 5, 7, 5, 8, 5, 8, 7, 10, 7, 9, 7, 10, 10, 12, 9, 13, 10, 14, 12, 14, 12, 15, 12, 16, 14, 17, 14, 18, 15, 20, 19, 23, 18, 21, 17, 22, 21, 24, 18, 22, 20, 26, 25, 28, 24, 29, 25, 31, 27, 29, 25, 30, 27
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OFFSET
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1,6
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COMMENTS
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Number of Goldbach partitions of [n-1, n-2, ..., n-floor(n/3)] into two parts whose smallest part is >= i, where i is the index in the list (i=1,2,..). For example, a(11)=4; The numbers 10, 9 and 8 contain a total of 4 Goldbach partitions into two parts whose smallest parts are greater than or equal to 1, 2, and 3 respectively. 10 = 7+3 = 5+5 (3,5 >= 1), 9 = 7+2 (2 >= 2), 8 = 5+3 (3 >= 3).
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LINKS
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FORMULA
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a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} A010051(i) * A010051(n-k-i).
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EXAMPLE
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Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
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n | 3 4 5 6 7 8 9 10 ...
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a(n) | 0 0 1 2 2 2 3 2 ...
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MATHEMATICA
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Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - k - i] - PrimePi[n - k - i - 1]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
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PROG
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(PARI) a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, ispseudoprime(i)*ispseudoprime(n-k-i))) \\ Felix Fröhlich, Apr 29 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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