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

0,8

Links

Table of n, a(n) for n=0..75.

Example

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.

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

Prog

(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

Crossrefs

Cf. A002375, A141095.

Sequence in context: A261036 A008612 A029320 * A187450 A187449 A102541

Adjacent sequences: A363989 A363990 A363991 * A363993 A363994 A363995

Keyword

nonn

Author

Brian Darrow, Jr., Jun 30 2023

Status

approved