A329586 Row lengths of A329585: number of solutions of the congruences x^2 == +1 (mod n) or (inclusive) x^2 == -1 (mod n), for n >= 1.

1, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 2, 8, 4, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 2, 4, 8, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 4, 2, 4, 8, 4, 4, 2, 8, 4, 2, 4, 4, 8, 4, 2, 4, 4, 4

Offset

1,2

Comments

See A329585 for details and examples (there n is called m).

For the number of all representative solutions z^2 = +1 (mod n), for n >= 1, with z = a + b*i, where a and b are nonnegative integers, see A227091.

Links

Table of n, a(n) for n=1..70.

Formula

a(1) = 1, a(2) = 2 (special case), and for n >= 3: a(n) = 2^{r2(e2) + r1 + r3} + delta_{r2(e2),0} * delta_{r3,0}*2^r1, where r1 = r1(n) and r3 = r3(n) are the number of the distinct odd primes congruent to 1 and 3 modulo 4, respectively, in the prime number factorization of n, and r2(e2) = 0, 1 and 2 if the power e2 of the even prime 2 is 0 (odd n case) or 1, 2 and >= 3, respectively, and delta is the Kronecker symbol. a(n) is always a nonnegative power of 2. (See A329585 for a sketch of the proof.)

Crossrefs

Cf. A329585, A329588, A227091.

Sequence in context: A366534 A066671 A159802 * A255336 A292929 A049627

Adjacent sequences: A329583 A329584 A329585 * A329587 A329588 A329589

Keyword

nonn,easy

Author

Wolfdieter Lang,Dec 14 2019

Status

approved