login
Number of Goldbach partitions of 2n into an odd-indexed prime and an even-indexed prime.
2

%I #6 Jun 04 2021 17:07:30

%S 0,0,0,1,0,1,1,0,2,1,0,3,1,0,3,0,2,2,1,1,2,2,1,3,2,1,4,0,1,5,1,1,4,1,

%T 3,2,2,4,3,2,1,5,3,2,3,2,3,4,0,3,6,2,3,1,5,7,0,4,4,5,2,3,4,2,5,2,3,4,

%U 4,4,2,5,4,4,3,3,4,7,1,3,7,3,2,5,5,3,7,2,3,8,3,2,9

%N Number of Goldbach partitions of 2n into an odd-indexed prime and an even-indexed prime.

%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..n} sign( ((pi(k)+1) mod 2) * (pi(2*n-k) mod 2) + (pi(k) mod 2) * ((pi(2*n-k)+1) mod 2) ) * c(k) * c(2*n-k), where c(n) is the prime characteristic.

%e a(12) = 3; 2*12 = 24 has 3 Goldbach partitions with one even-indexed prime and one odd-indexed prime: (19,5), (17,7) and (13,11). For example, 19 is the 8th prime and 5 is the 3rd prime, 17 is the 7th prime and 7 is the 4th prime, and 13 is the 6th prime while 11 is the 5th prime.

%t Table[Sum[Sign[Mod[PrimePi[k] + 1, 2] Mod[PrimePi[2 n - k], 2] + Mod[PrimePi[k], 2] Mod[PrimePi[2 n - k] + 1, 2]] (PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]), {k, n}], {n, 100}]

%Y Cf. A344981, A344982.

%K nonn

%O 1,9

%A _Wesley Ivan Hurt_, Jun 04 2021