A277776 Triangle T(n,k) in which the n-th row contains the increasing list of nontrivial square roots of unity mod n; n>=1.

3, 5, 5, 7, 4, 11, 7, 9, 9, 11, 8, 13, 5, 7, 11, 13, 17, 19, 13, 15, 11, 19, 15, 17, 10, 23, 6, 29, 17, 19, 14, 25, 9, 11, 19, 21, 29, 31, 13, 29, 21, 23, 19, 26, 7, 17, 23, 25, 31, 41, 16, 35, 25, 27, 21, 34, 13, 15, 27, 29, 41, 43, 20, 37, 11, 19, 29, 31, 41

Offset

1,1

Comments

Rows with indices n in A033948 (or with A046144(n)=0) are empty. Indices of nonempty rows are given by A033949.

This is A228179 without the trivial square roots {1, n-1}.

The number of terms in each nonempty row n is even: A060594(n)-2.

Links

Alois P. Heinz, Rows n = 1..5300, flattened

Example

Row n=8 contains 3 and 5 because 3*3 = 9 == 1 mod 8 and 5*5 = 25 == 1 mod 8.
Triangle T(n,k) begins:
08 :   3,  5;
12 :   5,  7;
15 :   4, 11;
16 :   7,  9;
20 :   9, 11;
21 :   8, 13;
24 :   5,  7, 11, 13, 17, 19;
28 :  13, 15;
30 :  11, 19;

Maple

T:= n-> seq(`if`(i*i mod n=1, i, [][]), i=2..n-2):
seq(T(n), n=1..100);
# second Maple program:
T:= n-> ({numtheory[rootsunity](2, n)} minus {1, n-1})[]:
seq(T(n), n=1..100);

Mathematica

T[n_] := Table[If[Mod[i^2, n] == 1, i, Nothing], {i, 2, n-2}];
Select[Array[T, 100], # != {}&] // Flatten (* Jean-François Alcover, Jun 18 2018, from first Maple program *)

Prog

(Python)
from itertools import chain, count, islice
from sympy.ntheory import sqrt_mod_iter
def A277776_gen(): # generator of terms
    return chain.from_iterable((sorted(filter(lambda m:1<m<n-1, sqrt_mod_iter(1, n))) for n in count(2)))
A277776_list = list(islice(A277776_gen(), 30)) # Chai Wah Wu, Oct 26 2022

Crossrefs

Columns k=1-2 give: A082568, A357099.

Last elements of nonempty rows give A277777.

Cf. A033948, A033949, A046144, A060594, A228179.

Sequence in context: A282624 A179858 A141501 * A103988 A285204 A086269

Adjacent sequences: A277773 A277774 A277775 * A277777 A277778 A277779

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Oct 30 2016

Status

approved