For the odd primes
p
1
=
3
,
{p_1} = 3,
,
p
2
=
5
,
⋯
,
{p_2} = 5, \cdots ,
we determine integer sequences
N
p
{N_p}
such that the Legendre symbol
(
N
/
N
p
i
p
i
)
=
±
1
({N \left / {\vphantom {N {{p_i}}}} \right . {{p_i}}}) = \pm 1
for all
p
i
≦
p
{p_i} \leqq p
for a prescribed array of signs
±
1
\pm 1
; (i.e., for a prescribed quadratic character). We examine six quadratic characters having special interest and applications. We present tables of these
N
p
{N_p}
and examine some applications, particularly to questions concerning extreme values for the smallest primitive root (of a prime
N
N
), the class number of the quadratic field
R
(
√
−
N
)
R(\surd - N)
, the real Dirichlet
L
L
functions, and quadratic character sums.