A352612 - OEIS

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

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OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..73.

EXAMPLE

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.

PROG

(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; }

CROSSREFS

Cf. A103131, A141098, A352587.

Sequence in context: A135680 A135682 A287415 * A083245 A111610 A119816

Adjacent sequences: A352609 A352610 A352611 * A352613 A352614 A352615

KEYWORD

nonn

AUTHOR

Craig J. Beisel, Mar 23 2022

STATUS

approved