A230366 -id:A230366 - cp's OEIS frontend

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

Darío Clavijo, May 27 2024

nonn

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

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

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)])

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

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

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

A373286 a(n) = Product_{k=1..n} (k^2 mod n if k^2 mod n > 0).

Original entry on oeis.org

1, 1, 1, 1, 16, 48, 64, 16, 784, 233280, 291600, 20736, 167961600, 281014272, 74649600, 1679616, 3229640294400, 2472693350400, 511642685030400, 2751882854400, 4854321355161600, 2085841207296000000, 216982220508758016, 15850845241344, 25975097361564696576

Offset: 1

Darío Clavijo, May 30 2024

nonn

a(n) mod A372651(n) = 0.

Cf. A165909, A230366, A372651.

a[n_]:=Product[Max[Mod[k^2,n],1],{k,n}]; Array[a,25] (* Stefano Spezia, May 30 2024 *)

a(n) = my(x); prod(k=1, n, if ((x=lift(Mod(k,n)^2))>0, x, 1)); \\ Michel Marcus, May 30 2024

def a(n):
  tmp = 1
  for i in range(1, n+1):
    if (k := pow(i,2,n)): tmp *= k
  return tmp
print([a(n) for n in range(1, 26)])

cp's OEIS Frontend

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

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A373286 a(n) = Product_{k=1..n} (k^2 mod n if k^2 mod n > 0).

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