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A344982
Number of Goldbach partitions of 2n into two odd-indexed primes.
2
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, 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, 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
OFFSET
1,11
FORMULA
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.
EXAMPLE
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).
MATHEMATICA
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}]
CROSSREFS
Cf. A000720 (pi), A010051, A344981 (even-indexed).
Sequence in context: A359288 A096142 A216921 * A359240 A280285 A033719
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 03 2021
STATUS
approved