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A344987
Number of Goldbach partitions of 2n into 2 primes where the smaller prime has an odd index and the larger prime has an even index.
1
0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 1, 1, 1, 1, 2, 1, 0, 1, 4, 0, 1, 1, 1, 3, 1, 0, 2, 3, 0, 2, 1, 1, 4, 0, 0, 4, 3, 0, 3, 1, 0, 4, 1, 0, 3, 3, 0, 2, 2, 1, 4, 1, 1, 3, 3, 0, 2, 4, 0, 1, 3, 1, 3, 4, 0, 2, 4, 1, 2, 4
OFFSET
1,12
FORMULA
a(n) = Sum_{k=1..n} (pi(k) mod 2) * ((pi(2*n-k)+1) mod 2) * c(k) * c(2*n-k), where c(n) is the prime characteristic.
a(n) = A342301(n) - A344986(n).
EXAMPLE
a(12) = 2; 2*12 = 24 has 2 Goldbach partitions where the smaller prime has an odd index and the larger prime has an even index: (19,5) and (13,11). For example, 19 is the 8th prime and 5 is the 3rd, while 13 is the 6th prime and 11 is the 5th.
MATHEMATICA
Table[Sum[Mod[PrimePi[k], 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
Sequence in context: A283669 A220945 A089224 * A359325 A325122 A234578
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 04 2021
STATUS
approved