A209389 - OEIS

%I #20 Aug 09 2022 14:15:53

%S 0,1,1,3,3,5,1,7,1,9,1,1,5,13,11,15,13,17,1,1,13,21,1,1,7,25,1,1,17,1,

%T 1,31,23,33,29,1,31,37,25,1,9,1,1,1,19,45,1,1,1,49,35,1,23,53,21,1,37,

%U 57,1,1,11,61,55,63,1,1,1,1,47,1

%N Product of positive odd integers smaller than n and relatively prime to n, taken Modd n. A209388(n) (Modd n).

%C For Modd n (not to be confused with mod n) see a comment on A203571.

%C See A209388 for the number of elements of the reduced residue class Modd n, called delta(n).

%C a(prime(n)) = (prime(n)-2)!! Modd prime(n) = 1 if n=1 or (prime(n)-1)/2 is odd, and = r(prime(n)) if (prime(n)-1)/2 is even. Here r(prime(n)) is the smallest positive nontrivial solution of x^2==1 (Modd prime(n)), which exists only for primes of the form 4*k+1 given in A002144. For r(prime(n)) see A206549. This is the analog of Wilson's theorem for Modd prime(n).

%C For (prime(n)-2)!! see A207332. [_Wolfdieter Lang_, Mar 28 2012]

%F a(n) = A209388(n) (Modd n), n>=1.

%e a(1) = 1 (Modd 1) = -1 (mod 1) = 0, because floor(1/1)=1 is odd. a(4)= 1*3 (Modd 4) = 3, a(15) = 1*7*11*13 (Modd 15) = 1001 (Modd 15) = 1001 (mod 15) because floor(1001/15) = 66 is even, hence a(15) = 11.

%Y Cf. A209388, A160377 (mod n analog).

%K nonn,easy

%O 1,4

%A _Wolfdieter Lang_, Mar 10 2012