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A234537
Number of nontrivial non-Goldbach partitions of 2n into two odd parts (with smaller part greater than 1).
1
0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 5, 4, 4, 7, 6, 6, 7, 7, 6, 8, 9, 8, 10, 10, 8, 12, 10, 10, 14, 12, 11, 13, 13, 12, 15, 15, 12, 16, 17, 13, 18, 18, 16, 21, 18, 17, 20, 20, 18, 21, 20, 18, 22, 23, 17, 26, 25, 21, 28, 25, 23, 27, 28, 26, 27, 27, 24
OFFSET
1,9
COMMENTS
Number of partitions of 2n into two odd parts with at least 1 composite part less than 2n-1.
FORMULA
a(n) = ceiling(n/2) - 1 - Sum_{i=3..n} A010051(i) * A010051(2n-i).
EXAMPLE
a(15) = 4; there are exactly 4 partitions of 2*15 = 30 into two odd parts with at least one composite part less than 2*15 - 1 = 29: (27,3), (25,5), (21,9), (15,15).
MATHEMATICA
Table[Ceiling[n/2] - 1 - Sum[(PrimePi[i] - PrimePi[i - 1])*(PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, 3, n}], {n, 100}]
PROG
(PARI) a(n)=my(s); forstep(k=3, n, 2, if(!isprime(k) || !isprime(2*n-k), s++)); s \\ Charles R Greathouse IV, Jul 30 2016
(Python)
from sympy import isprime
def a(n): return sum(1 for k in range(3, n + 1, 2) if not isprime(k) or not isprime(2*n - k))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017
CROSSREFS
Sequence in context: A103372 A029082 A035450 * A029126 A269850 A036054
KEYWORD
nonn,nice
AUTHOR
Wesley Ivan Hurt, Dec 27 2013
STATUS
approved