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A334152
Number of Goldbach partitions (p,q) of 2n, such that p-2 and q-2 are both prime or both not prime.
1
0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 1, 1, 3, 1, 0, 2, 2, 1, 2, 3, 0, 2, 2, 2, 3, 2, 1, 2, 3, 1, 3, 3, 2, 1, 5, 0, 4, 3, 3, 2, 5, 2, 3, 3, 3, 3, 4, 3, 0, 6, 1, 4, 4, 5, 2, 5, 3, 5, 5, 4, 3, 5, 4, 1, 7, 1, 5, 4, 5, 2, 6, 4, 4, 5, 5, 3, 6, 4, 3, 8, 1, 4, 6, 6, 4, 5, 5, 3, 6
OFFSET
1,7
FORMULA
a(n) = Sum_{i=1..n} [c(i-2) = c(2*n-i-2)] * c(i) * c(2*n-i), where [] is the Iverson bracket and c is the prime characteristic (A010051).
EXAMPLE
a(6) = 1; 2*6 = 12 has one Goldbach partition, (7,5), such that 7-2 = 5 and 5-2 = 3 (both prime).
a(7) = 2; 2*7 = 14 has two Goldbach partitions, (11,3) and (7,7), such that 11-2 = 9 and 3-2 = 1 (both not prime) as well as 7-2 = 5 (both prime).
MATHEMATICA
Table[Sum[KroneckerDelta[PrimePi[i - 2] - PrimePi[i - 3], PrimePi[2 n - i - 2] - PrimePi[2 n - i - 3]]*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 100}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 16 2020
STATUS
approved