A372651 a(n) is the product of the distinct nonzero quadratic residues of n.

1, 1, 1, 1, 4, 12, 8, 4, 28, 1080, 540, 36, 12960, 44352, 2160, 36, 1797120, 524160, 22619520, 2880, 1088640, 4790016000, 465813504, 6912, 5096577024, 8115883776000, 5477472000, 2419200, 267346759680000, 124104960000, 216218419200000, 244800, 143187264000

Offset: 1

Author

Darío Clavijo, May 27 2024

Keywords

• nonn

Crossrefs

Cf. A046071, A105612, A165909, A230366, A232195.

Programs

PARI

a(n) = my(list=List()); for (i=1, n-1, if (issquare(Mod(i, n)), listput(list, i))); vecprod(Vec(list)); \\ Michel Marcus, May 28 2024

Python

from sympy import prod
def a(n):
  k, QS = 0,[]
  for i in range((n >> 1) + 1):
    if k > 0: QS.append(k)
    k += (i << 1) + 1
    k %= n
  return prod(set(QS))
print([a(n) for n in range(1, 34)])

Python

from math import prod
from sympy.ntheory.residue_ntheory import quadratic_residues
def A372651(n): return prod(r for r in quadratic_residues(n) if r) # Chai Wah Wu, May 30 2024

Formula

a(n) mod n = A232195(n).

a(n) = Product_{k=1..n} A046071(n,k).