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A344986
Number of Goldbach partitions of 2n into 2 primes where the smaller prime has an even-index and the larger prime has an odd index.
1
0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 0, 0, 3, 1, 0, 2, 0, 2, 1, 1, 2, 2, 2, 0, 1, 3, 1, 2, 1, 0, 3, 0, 1, 3, 2, 1, 0, 4, 3, 0, 4, 0, 2, 2, 0, 3, 2, 1, 1, 3, 1, 1, 4, 0, 3, 3, 0, 2, 2, 1, 4, 1, 1, 3, 3, 1, 2, 4, 0, 3, 2, 1, 4, 2, 0, 5
OFFSET
1,15
FORMULA
a(n) = Sum_{k=1..n} ((pi(k)+1) mod 2) * (pi(2*n-k) mod 2) * c(k) * c(2*n-k), where c(n) is the prime characteristic.
a(n) = A342301(n) - A344987(n).
EXAMPLE
a(15) = 2; There are 2 Goldbach partitions of 2*15 = 30 where the smaller part is an even-indexed prime and the larger part is an odd-indexed prime: (23,7) and (17,13). For example, 23 is the 9th prime and 7 is the 4th prime, while 17 is the 7th prime and 13 is the 6th prime.
MATHEMATICA
Table[Sum[Mod[PrimePi[k] + 1, 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
Sequence in context: A105553 A342026 A262696 * A117165 A213629 A357380
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 04 2021
STATUS
approved