A344982 - OEIS

%I #5 Feb 16 2025 08:34:02

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

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

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

%N Number of Goldbach partitions of 2n into two odd-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) mod 2) * (pi(2*n-k) mod 2) * c(k) * c(2*n-k), where c(n) is the prime characteristic.

%e a(11) = 2; There are 2 Goldbach partitions of 2*11 = 22 into two odd-indexed primes: (17,5) and (11,11) where 5, 11, and 17 are primes with odd indices (i.e., 5 is the 3rd prime, 11 is the 5th prime and 17 is the 7th prime).

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

%Y Cf. A000720 (pi), A010051, A344981 (even-indexed).

%K nonn

%O 1,11

%A _Wesley Ivan Hurt_, Jun 03 2021