A334207 Number of ways to write 2n as the sum of two nonprime positive integers.

1, 0, 0, 1, 2, 2, 2, 4, 4, 4, 5, 6, 6, 7, 8, 7, 9, 11, 8, 11, 12, 11, 12, 14, 14, 14, 16, 15, 16, 19, 15, 19, 21, 17, 21, 22, 20, 22, 25, 22, 23, 27, 24, 25, 30, 26, 27, 31, 27, 31, 33, 30, 31, 34, 32, 34, 37, 34, 34, 42, 34, 37, 43, 36, 41, 43, 40, 41, 44, 43

Offset: 1

Author

Wesley Ivan Hurt, Apr 18 2020

Keywords

• nonn
• easy

Examples

			a(8) = 4; 2*8 = 16 has four partitions into nonprime parts, (15,1), (12,4), (10,6) and (8,8).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051.

Bisection of A062610.

Programs

Maple

N:= 1000: # for a(1)..a(N)
NP:= remove(isprime, [$1..2*N]):
V:= Vector(N):
for i from 1 to nops(NP) do
  for j from i do
    x:= NP[i]+NP[j];
    if x > 2*N then break fi;
    if x::even then
      V[x/2]:= V[x/2]+1;
    fi
od od:
convert(V,list); # Robert Israel, Apr 20 2020

Mathematica

Table[Sum[(1 - PrimePi[i] + PrimePi[i - 1]) (1 - PrimePi[2 n - i] + PrimePi[2 n - i - 1]), {i, n}], {n, 100}]

Formula

a(n) = Sum_{i=1..n} (1 - c(i)) * (1 - c(2*n - i)), where c is the prime characteristic (A010051).