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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).

%I A352612 #30 Mar 24 2022 15:07:41
%S A352612 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,
%T A352612 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,
%U A352612 51,55,1,49,58,1,4,37,59,55,1,49,66,33,65,31,70,25,4
%N 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).
%C A352612 The convention for the empty product here is 1. The second product exists for all numbers greater than 210. See A141098.
%C A352612 Conjecture: For odd n, if a(n) == -1 (mod n) then n must be a prime power.
%e A352612 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.
%e A352612 For n=25 these products exist and are given -44618574^2*12096 == 4 (mod 25). Therefore, a(25)=4.
%o A352612 (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; }
%Y A352612 Cf. A103131, A141098, A352587.
%K A352612 nonn
%O A352612 1,3
%A A352612 _Craig J. Beisel_, Mar 23 2022