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%I #13 Jun 15 2020 23:34:35
%S 0,0,0,1,0,1,0,1,2,1,0,1,0,1,2,1,0,1,0,1,2,1,0,1,2,3,2,1,0,1,0,1,2,3,
%T 2,1,0,1,2,1,0,1,0,1,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,0,1,2,3,2,1,0,1,
%U 2,1,0,1,0,1,2,3,2,1,0,1,2,1,0,1,2,3,2,1,0
%N Number of partitions of 2n into two composite parts, (r,s), such that r and s have the same number of primes less than or equal to them.
%C Apparently a(n) = A051699(n) for n>=2. - _R. J. Mathar_, Apr 22 2020
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=2..n} [pi(i) = pi(2*n-i)] * (1 - c(i)) * (1 - c(2*n-i)), where [] is the Iverson bracket, pi is the prime counting function (A000720), and c is the prime characteristic (A010051).
%e a(9) = 2; 2*9 = 18 has two partitions into composite parts, (10,8) and (9,9), such that pi(10) = 4 = pi(8) and pi(9) = 4 = pi(9).
%t Table[Sum[KroneckerDelta[PrimePi[i], PrimePi[2 n - i]] (1 - PrimePi[i] + PrimePi[i - 1]) (1 - PrimePi[2 n - i] + PrimePi[2 n - i - 1]), {i, 2, n}], {n, 100}]
%Y Cf. A000720, A010051, A051699.
%K nonn,easy
%O 1,9
%A _Wesley Ivan Hurt_, Apr 18 2020
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