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%I #17 Feb 13 2020 05:39:09
%S 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,
%T 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,
%U 4,4
%N 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.
%C See A329585 for details and examples (there n is called m).
%C 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.
%F 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.)
%Y Cf. A329585, A329588, A227091.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_,Dec 14 2019
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