A277776 -id:A277776 - cp's OEIS frontend

A082568 First nontrivial square root of unity mod A033949(n), i.e., smallest x > 1 such that x^2 == 1 mod A033949(n).

3, 5, 4, 7, 9, 8, 5, 13, 11, 15, 10, 6, 17, 14, 9, 13, 21, 19, 7, 16, 25, 21, 13, 20, 11, 8, 31, 14, 23, 33, 22, 29, 17, 26, 37, 34, 25, 9, 13, 16, 28, 21, 19, 27, 45, 32, 39, 17, 10, 49, 35, 25, 29, 53, 21, 38, 15, 37, 24, 57, 53, 50, 11, 40, 61, 55, 63, 44

Offset: 1

Jon Perry, May 06 2003

nonn

			a(3) = 4 because A033949(3) = 15 and 4^2 = 16 == 1 mod 15 is the first integer to do so.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A033949.

Column k=1 of A277776.

for (n=3,100, for (j=2,n-2,if (j^2%n==1,print1(j","); break)))

from itertools import chain, count, islice
from sympy.ntheory import sqrt_mod_iter
def A082568_gen(): # generator of terms
    return chain.from_iterable((sorted(filter(lambda m:1A082568_list = list(islice(A082568_gen(),30)) # Chai Wah Wu, Oct 26 2022

Offset corrected, name clarified and more terms from Alois P. Heinz, Oct 30 2016

A228179 Irregular table where the n-th row consists of the square roots of 1 in Z_n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 3, 5, 7, 1, 8, 1, 9, 1, 10, 1, 5, 7, 11, 1, 12, 1, 13, 1, 4, 11, 14, 1, 7, 9, 15, 1, 16, 1, 17, 1, 18, 1, 9, 11, 19, 1, 8, 13, 20, 1, 21, 1, 22, 1, 5, 7, 11, 13, 17, 19, 23, 1, 24, 1, 25, 1, 26, 1, 13, 15, 27, 1, 28, 1, 11

Offset: 2

Tom Edgar, Aug 20 2013

Each 1 starts a new row.
This is a subsequence of A020652.
Row n has A060594(n) entries.
Each row forms a subgroup of the multiplicative group of units of Z_n.

			The table starts out as follows:
  1
  1 2
  1 3
  1 4
  1 5
  1 6
  1 3 5 7
  1 8
  1 9
  1 10
  1 5 7 11
  ...

Alois P. Heinz, Rows n = 2..2000 of irregular triangle, flattened

Cf. A060594, A038566, A020652, A082568.
Cf. A070667 (second column), A358016 (second-last column).
Cf. A277776 (nontrivial square roots of 1).

T:= n-> seq(`if`(k&^2 mod n=1, k, NULL), k=1..n-1):
seq(T(n), n=2..50);  # Alois P. Heinz, Aug 20 2013

Flatten[Table[Position[Mod[Range[n]^2, n], 1], {n, 2, 50}]] (* T. D. Noe, Aug 20 2013 *)

from itertools import chain, count, islice
from sympy.ntheory import sqrt_mod_iter
def A228179_gen(): # generator of terms
    return chain.from_iterable((sorted(sqrt_mod_iter(1,n)) for n in count(2)))
A228179_list = list(islice(A228179_gen(),30)) # Chai Wah Wu, Oct 26 2022

[[i for i in [1..k-1] if (i*i).mod(k)==1] for k in [2..n]] #changing n gives you the table up to the n-th row.

A277777 Largest nontrivial square root of unity modulo the n-th positive integer that does not have a primitive root (A033949).

Original entry on oeis.org

5, 7, 11, 9, 11, 13, 19, 15, 19, 17, 23, 29, 19, 25, 31, 29, 23, 26, 41, 35, 27, 34, 43, 37, 49, 55, 33, 51, 43, 35, 47, 41, 55, 49, 39, 43, 53, 71, 71, 69, 59, 67, 71, 64, 47, 61, 56, 79, 89, 51, 67, 79, 76, 55, 89, 73, 97, 77, 91, 59, 64, 69, 109, 83, 63, 71

Offset: 1

Alois P. Heinz, Oct 30 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Root of unity modulo n

Last elements of nonempty rows of A277776.

Cf. A033948, A033949, A082568.

from gmpy2 import *
def f(n):
  for k in range(n - 2, 0, -1):
    if pow(k, 2, n) == 1:
      return k
def A277777(L):
  return [j for j in [f(k) for k in range(3, L + 1)] if j > 1] # Darío Clavijo, Oct 15 2022

from itertools import count, islice
from sympy.ntheory import sqrt_mod_iter
def A277777_gen(): # generator of terms
    for n in count(3):
        if (m:=max(filter(lambda k:k 1:
            yield m
A277777_list = list(islice(A277777_gen(),30)) # Chai Wah Wu, Oct 26 2022

a(n) = A033949(n) - A082568(n).

A357099 Second nontrivial square root of unity mod A033949(n), i.e., second smallest x > 1 such that x^2 == 1 mod the n-th positive integer that does not have a primitive root.

Original entry on oeis.org

5, 7, 11, 9, 11, 13, 7, 15, 19, 17, 23, 29, 19, 25, 11, 29, 23, 26, 17, 35, 27, 34, 15, 37, 19, 55, 33, 51, 43, 35, 47, 41, 19, 49, 39, 43, 53, 31, 29, 69, 59, 23, 71, 64, 47, 61, 56, 31, 89, 51, 67, 27, 34, 55, 89, 73, 41, 77, 91, 59, 64, 69, 19, 83, 63, 71

Offset: 1

Alois P. Heinz, Oct 25 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=2 of A277776.

Cf. A033949, A082568.

from itertools import count, chain, islice
from sympy.ntheory import sqrt_mod_iter
def A357099_gen(): # generator of terms
    return chain.from_iterable((sorted(filter(lambda m:1A357099_list = list(islice(A357099_gen(),30)) # Chai Wah Wu, Oct 26 2022

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A082568 First nontrivial square root of unity mod A033949(n), i.e., smallest x > 1 such that x^2 == 1 mod A033949(n).

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A228179 Irregular table where the n-th row consists of the square roots of 1 in Z_n.

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A277777 Largest nontrivial square root of unity modulo the n-th positive integer that does not have a primitive root (A033949).

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A357099 Second nontrivial square root of unity mod A033949(n), i.e., second smallest x > 1 such that x^2 == 1 mod the n-th positive integer that does not have a primitive root.

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Python