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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.
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%I #15 Aug 02 2023 12:02:18

%S 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,

%T 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,

%U 3,22,18,7,24,14,11,20,20,14,17,18,10,22,22,8

%N 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.

%e 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.

%p f:= proc(n) local k;

%p 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)]))

%p end proc:

%p map(f, [$0..100]); # _Robert Israel_, Jul 03 2023

%o (Sage)

%o def d(a):

%o """

%o 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

%o """

%o d=0

%o for i in range(1,a+1):

%o 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)):

%o d=d+1

%o return d

%Y Cf. A002375, A141095.

%K nonn

%O 0,8

%A _Brian Darrow, Jr._, Jun 30 2023