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

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

Author

Alois P. Heinz, Oct 30 2016

Keywords

• nonn
• look

Links

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

Crossrefs

Last elements of nonempty rows of A277776.

Cf. A033948, A033949, A082568.

Programs

Python

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

Python

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

Formula

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