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A226237
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Sum of the parts in the Goldbach partitions of 2n.
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4
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0, 4, 6, 8, 20, 12, 28, 32, 36, 40, 66, 72, 78, 56, 90, 64, 136, 144, 76, 120, 168, 132, 184, 240, 200, 156, 270, 168, 232, 360, 186, 320, 396, 136, 350, 432, 370, 380, 546, 320, 410, 672, 430, 352, 810, 368, 470, 672, 294, 600, 816, 520, 636, 864, 660, 784
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OFFSET
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1,2
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COMMENTS
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Goldbach's Conjecture states that every positive even integer > 4 is expressible as the sum of two odd primes in at least one way. This is logically equivalent to the statement that a(n) > 0 for n > 2.
The sum of the parts in the partitions of 2n into exactly two prime parts.
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LINKS
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Table of n, a(n) for n=1..56.
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions
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FORMULA
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a(n) = 2n * A045917(n). a(n) = A185297(n) + A187129(n), n>1.
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EXAMPLE
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a(13) = 78. Since 2*13 = 26 has exactly 3 Goldbach partitions: (23,3),(19,7), and (13,13). The sum of the parts gives: 23+19+13+13+7+3 = 78.
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MAPLE
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with(numtheory); A226237:=n->2*n*sum( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)), i=1..n); seq(A226237(n), n=1..100);
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MATHEMATICA
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Table[ 2 n*Sum[ Floor[2/PrimeOmega[2 n*i - i^2]], {i, 2, n}], {n,
100}]
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CROSSREFS
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Cf. A045917, A185297, A187129, A187619 (Sum of differences).
Sequence in context: A346868 A279896 A247280 * A114315 A058238 A191697
Adjacent sequences: A226234 A226235 A226236 * A226238 A226239 A226240
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KEYWORD
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nonn,easy
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AUTHOR
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Wesley Ivan Hurt, Aug 25 2013
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STATUS
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approved
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