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A344981
Number of Goldbach partitions of 2n into two even-indexed primes.
2
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, 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, 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
OFFSET
1,13
FORMULA
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.
EXAMPLE
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).
MATHEMATICA
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}]
CROSSREFS
Cf. A000720 (pi), A010051, A344982 (odd-indexed).
Sequence in context: A324730 A118683 A175800 * A161116 A262726 A112605
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 03 2021
STATUS
approved