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Number of Goldbach partitions of 2n into two even-indexed primes.
2

%I #7 Feb 16 2025 08:34:02

%S 0,0,1,0,1,0,1,1,0,1,1,0,2,0,0,2,0,1,1,1,1,1,1,1,2,0,0,3,1,1,1,1,2,1,

%T 0,2,3,0,1,2,2,1,2,0,3,2,0,2,2,1,1,2,1,4,1,0,4,2,1,3,2,1,3,0,1,4,1,1,

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

%N Number of Goldbach partitions of 2n into two even-indexed primes.

%H Eric Weisstein's World of Mathematics, <a href="https://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} ((pi(k)+1) mod 2) * ((pi(2*n-k)+1) mod 2) * c(k) * c(2*n-k), where c(n) is the prime characteristic.

%e a(13) = 2; There are 2 Goldbach partitions of 2*13 = 26 into two even-indexed primes: (19,7) and (13,13) where 7, 13, and 19 are primes with even indices (i.e., 7 is the 4th prime, 13 is the 6th prime and 19 is the 8th prime).

%t Table[Sum[Mod[PrimePi[k] + 1, 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. A000720 (pi), A010051, A344982 (odd-indexed).

%K nonn,changed

%O 1,13

%A _Wesley Ivan Hurt_, Jun 03 2021