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

1,9

Comments

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

Links

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

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

Cf. A002375, A141100.

Sequence in context: A103372 A029082 A035450 * A029126 A269850 A036054

Adjacent sequences: A234534 A234535 A234536 * A234538 A234539 A234540

Keyword

nonn,nice

Author

Wesley Ivan Hurt, Dec 27 2013

Status

approved