A207332 -id:A207332 - cp's OEIS frontend

A209388 Product of positive odd integers smaller than n and relatively prime to n.

1, 1, 1, 3, 3, 5, 15, 105, 35, 189, 945, 385, 10395, 19305, 1001, 2027025, 2027025, 85085, 34459425, 8729721, 230945, 1249937325, 13749310575, 37182145, 4216455243, 608142583125, 929553625, 1452095555625, 213458046676875, 215656441, 6190283353629375

Offset: 1

Wolfdieter Lang, Mar 10 2012

This is the product over the smallest positive representatives of the odd reduced residue class Modd n. For Modd n (not to be confused with mod n) see a comment on A203571. This reduced residue class has delta(n)=A055034(n) members.

The Moddn values of this sequence are given in A209339.

			a(4) = 1*3 = 3.
a(5) = 1*3 = 3.
a(15) = 1*7*11*13 = 1001.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A001783 (mod n analog), A207332, A209339.

Table[Times @@ Select[Range[1, n, 2], GCD[n, #] == 1 &], {n, 40}] (* T. D. Noe, Mar 12 2012 *)

a(n) = prod(k=1, n, if (k % 2, k, 1)); \\ Michel Marcus, Mar 12 2022

a(n) = product(2*k+1, k from {0,1,...,floor((n-2)/2)} and gcd(2*k+1,n) =1). a(1):=1 (empty product).
a(n) = product(k, k from {1,...,n-1} and gcd(k,2*n) = 1). a(1):=1 (empty product).
a(prime(n)) = (prime(n)-2)!! = A207332(n), for primes prime(n)=A000040(n).

A209389 Product of positive odd integers smaller than n and relatively prime to n, taken Modd n. A209388(n) (Modd n).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Mar 10 2012

For Modd n (not to be confused with mod n) see a comment on A203571.
See A209388 for the number of elements of the reduced residue class Modd n, called delta(n).
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).
For (prime(n)-2)!! see A207332. [Wolfdieter Lang, Mar 28 2012]

			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.

Cf. A209388, A160377 (mod n analog).

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

cp's OEIS Frontend

A209388 Product of positive odd integers smaller than n and relatively prime to n.

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