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%I #21 Sep 07 2019 08:16:56
%S 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,
%T 12,9,13,10,14,12,14,12,15,12,16,14,17,14,18,15,20,19,23,18,21,17,22,
%U 21,24,18,22,20,26,25,28,24,29,25,31,27,29,25,30,27
%N Number of partitions of n into 3 parts such that the middle and largest parts are both prime.
%C 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).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} A010051(i) * A010051(n-k-i).
%e Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e 1+1+8
%e 1+1+7 1+2+7
%e 1+2+6 1+3+6
%e 1+1+6 1+3+5 1+4+5
%e 1+1+5 1+2+5 1+4+4 2+2+6
%e 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e 1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
%e -----------------------------------------------------------------------
%e n | 3 4 5 6 7 8 9 10 ...
%e -----------------------------------------------------------------------
%e a(n) | 0 0 1 2 2 2 3 2 ...
%e -----------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 07 2019
%t 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}]
%o (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
%Y Cf. A010051.
%K nonn,easy
%O 1,6
%A _Wesley Ivan Hurt_, Apr 27 2019
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