A352612 (n-1)*prod(-p^2 where 2 <= p <= n-2 is prime and relatively prime to n)*prod(k where both k and (n-k) are composite and relatively prime to n) (mod n).

0, 1, 2, 3, 4, 5, 4, 7, 4, 9, 10, 11, 4, 9, 11, 1, 16, 17, 4, 1, 17, 17, 22, 23, 4, 1, 26, 1, 28, 29, 4, 17, 29, 25, 1, 25, 36, 11, 35, 39, 40, 25, 4, 7, 41, 27, 46, 23, 4, 31, 1, 23, 52, 1, 51, 55, 1, 49, 58, 1, 4, 37, 59, 55, 1, 49, 66, 33, 65, 31, 70, 25, 4

Offset: 1

Author

Craig J. Beisel, Mar 23 2022

Keywords

• nonn

Comments

The convention for the empty product here is 1. The second product exists for all numbers greater than 210. See A141098.

Conjecture: For odd n, if a(n) == -1 (mod n) then n must be a prime power.

Examples

			For n=6 there are no prime totatives between 2 and 4 and there are also no composite totative pairs which add to 6 so both products do not exist and a(6)=n-1=5.
For n=25 these products exist and are given -44618574^2*12096 == 4 (mod 25). Therefore, a(25)=4.
		

Crossrefs

Cf. A103131, A141098, A352587.

Programs

PARI

a(n)= {prod_p=1; prod_r=1; for(k=2, n-2, if(gcd(k,n)==1, if(isprime(k), prod_p=prod_p*k*(n-k); ); if(!isprime(k) && !isprime(n-k), prod_r=prod_r*k; );); ); (-prod_p*prod_r)%n; }