%I #15 Sep 29 2020 17:03:27

%S 2,2,4,4,4,2,4,4,2,12,2,8,4,4,12,4,4,4,4,2,4,24,4,4,4,2,8,4,4,12,2,4,

%T 8,4,24,2,4,8,4,12,12,8,4,4,4,12,24,4,8,8,2,4,8,12,4,4,2,4,8,2,12,12,

%U 24,8,4,4,12,4,8,4,4,12,4,2,4,48

%N Number of representative solutions (a, b) of the complex congruence z^2 == +1 (mod m) with z = a + b*i and nonvanishing a*b.

%C This sequence gives one half of the row lengths of the irregular triangle A329587.

%C For the number of representative solutions of this congruence for all positive moduli m and vanishing a*b see A329586.

%C The formula for the number of representative solutions a(n), with modulus m = m(n), given in A329587, can be found from the number of all such solutions for m = m(n) given in A227091 after subtracting the number of solutions with a*b = 0 given in A329586. For odd m = m(n) this is S(m) = 2^(r1(m) +r3(m))*(2^r1(m) - 1) - delta(r3(m), 0)*2^(r1(m)), with r1(m) and r3(m) the number of distinct primes 1 (mod 4) and 3 (mod 4) in the prime number factorization of m respectively, and delta is the Kronecker symbol. For even m this is S(m) = 2^(r1(m) + r3(m))*(2^(1+r1(m)) - 1) - delta(r3(m), 0)*2^(r1(m)) if m/2 is odd (e2 = 1), and otherwise S(m) = 2^(r2(e2(m)) + r1(m) + r3(m))*(2^(1 + r1(m) + r3(m)) - 1), with r2(e2(m)) = 1 or 2 if e2(m) = 2 or >= 3, if m/2^(e2(m)) is odd.

%F a(n) is the number of solutions (a, b) of the congruence z^2 == +1 (mod m(n)), with z = a + b*i and a*b not equal to 0, for n >= 1. For m = m(n) see A329587: it is the sequence of even numbers >= 4 combined with the odd numbers from A329589, sorted increasingly.

%e n = 1, m = 4: a(1) = S(4) = 2^1 *(2^1 - 1) = 2.

%e n = 2, m = 6 = 2*3: a(2) = S(6) = 2^(0+1)*(2^1 - 1) - 0 = 2.

%e n = 3, m = 8 = 2^3: a(3) = S(8) = 2^2*(2^1 - 1) = 4.

%e n = 4, m = 10 = 2*5: a(4) = S(10) = 2^(1+0)*(2^(1+1) -1) - 2^1 = 2*3 - 2 = 4.

%e n = 10, m = 20 = 2^2*5: a(10) = S(20) = 2^(1+1+0)*(2^(1+1+0) - 1) = 4*3 = 12.

%e n = 15, m = 30 = 2*3*5: a(15) = S(30) = 2^(1+1)*(2^(1+1) - 1) - 0 = 4*3 = 12.

%Y Cf. A329585, A329586, A329587, A329589.

%K nonn

%O 1,1

%A _Wolfdieter Lang_, Dec 14 2019