A363992 The number of ways 2n can be expressed as the sum of an odd prime number and an odd nonprime, both of which are relatively prime to n.

0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 3, 3, 1, 6, 3, 1, 8, 4, 2, 6, 6, 3, 5, 7, 4, 8, 8, 2, 12, 7, 3, 13, 6, 6, 11, 9, 4, 12, 12, 4, 13, 13, 3, 14, 14, 8, 17, 11, 7, 15, 15, 10, 14, 13, 7, 16, 18, 3, 22, 18, 7, 24, 14, 11, 20, 20, 14, 17, 18, 10, 22, 22, 8

Offset: 0

Author

Brian Darrow, Jr., Jun 30 2023

Keywords

• nonn

Examples

			For n=24 (2n=48), we have a(24)=3 since 48=1+47, 48=13+35, and 48=23+25. These are the only sums containing one prime and one nonprime, both of which are relatively prime to n.
		

Crossrefs

Cf. A002375, A141095.

Programs

Maple

f:= proc(n) local k;
   nops(select(k -> igcd(n,k) = 1 and igcd(n,2*n-k) = 1 and isprime(k) and not isprime(2*n-k), [seq(k,k=1..2*n-1,2)]))
end proc:
map(f, [$0..100]); # Robert Israel, Jul 03 2023

Sage

def d(a):
    """
    This function returns the number of ways n=2a can be expressed as the sum of one prime number and an odd composite that are relatively prime to n
    """
    d=0
    for i in range(1,a+1):
        if ((is_prime(i) and not is_prime(2*a-i) and gcd(i,2*a-i) == 1)) or ((not is_prime(i) and is_prime(2*a-i) and gcd(i,2*a-i) == 1)):
            d=d+1
    return d