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

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

Author

Tom Edgar, Aug 20 2013

Keywords

• nonn
• easy
• tabf

Comments

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.

Examples

			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
  ...
		

Links

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

Crossrefs

Cf. A060594, A038566, A020652, A082568.

Cf. A070667 (second column), A358016 (second-last column).

Cf. A277776 (nontrivial square roots of 1).

Programs

Maple

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

Mathematica

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

Python

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

Sage

[[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.