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A334158
Number of Goldbach partitions (p,q) of 2n such that primes p and q can be written as the sum of two primes.
0
0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 1, 3, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 3, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 3, 0, 1, 1, 0, 0, 2, 0, 1, 3
OFFSET
1,13
FORMULA
a(n) = Sum_{i=1..n} c(i) * c(i-2) * c(2*n-i) * c(2*n-i-2), where c is the prime characteristic (A010051).
EXAMPLE
a(5) = 1; 2*5 = 10 has one Goldbach partition, (5,5), and 5 = 3 + 2 (both prime).
a(13) = 2; 2*13 = 26 has two Goldbach partitions, (19,7) and (13,13), and 19 = 17 + 2 (primes), 7 = 5 + 2 (primes) and 13 = 11 + 2 (primes).
MATHEMATICA
Table[Sum[(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
Cf. A010051.
Sequence in context: A287066 A324816 A321448 * A116357 A035168 A255647
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 16 2020
STATUS
approved