A234537 Number of nontrivial non-Goldbach partitions of 2n into two odd parts (with smaller part greater than 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

Author

Wesley Ivan Hurt, Dec 27 2013

Keywords

• nonn
• nice

Comments

Number of partitions of 2n into two odd parts with at least 1 composite part less than 2n-1.

Examples

			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).
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A002375, A141100.

Programs

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}]

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

Formula

a(n) = ceiling(n/2) - 1 - Sum_{i=3..n} A010051(i) * A010051(2n-i).