cp's OEIS Frontend

Note that {Kronecker(D,i)} is a Dirichlet character mod |D| if and only if D == 0, 1 (mod 4).

From the comment in A329648, D is a term if and only if there exists a prime p being a factor of D such that if we write D = p^e * s, gcd(p,s) = 1, then e is even and Kronecker(-s,p) = 1; if p = 2, then s == 7 (mod 8).

If D is a term, then so is D*k^2 for any k. The "primitive" terms are given by A329307.

Let S be the set of the positive integers congruent to 0 or 3 mod 4, S_2 = {2^e * s: e is even, e > 0, s == 7 (mod 8)}, S_p = {p^e * s: e is even, e > 0, s is in S, Kronecker(-s,p) = 1} for odd primes p, then S_p has density 1/(2p*(p+1)) over S; for any x in S, "x is in S_2", "x is in S_3", "x is in S_5", ... are mutually independent. This sequence is Union_{prime p} S_p, so this sequence has density 1 - Product_{primes p} (1 - 1/(2p*(p+1))) ~ 0.156234 over S.

Note that {Kronecker(D,i)} is a Dirichlet character mod |D| if and only if D == 0, 1 (mod 4).

Primitive terms in A329306.

Numbers of the form d*p^2, where -d is a fundamental discriminant and Kronecker(-d,p) = 1 (i.e., the rational prime p decomposes in the quadratic number field with discriminant -d).