

A307836


Number of Heegner rings in which prime(n) splits minus the number of Heegner rings in which it stays inert.


0



4, 4, 3, 4, 0, 3, 1, 2, 1, 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 5, 1, 1, 1
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OFFSET

1,1


COMMENTS

Given the nine quadratic integer rings with complex numbers which are also unique factorization domains, it stands to reason that most primes in Z remain inert in some of these quadratic rings and split in others.
So if we add up Legendre(H_i, p), where H_i iterates over the Heegner numbers (A003173 multiplied by 1) and p is an odd prime, we will generally find this sum to be odd and greater than 9 but less than 9.
Indeed the first occurrence of a(n) = 9 corresponds to the prime 3167, and the first occurrence of a(n) = 9 corresponds to the prime 15073. The first occurrence of a(n) = 7 or 7 corresponds to the prime 709.
The only even values in this sequence correspond the primes of A003173 (all nine except for 1). The only 0 corresponds to the prime 11; the three instances of 4 correspond to the primes 2, 3, 7; one instance of 2 for 19; two instances of 2 for 43 and 67; and one instance of 4 for 163.
Although (1 + i) is a ramifying ideal in Z[i], Kronecker(1, 2) = 1, so here 2 is counted as splitting (+1) rather than ramifying (0) in Z[i].


LINKS

Table of n, a(n) for n=1..81.


FORMULA

a(n) = Sum_{i = 1..9} Kronecker(H_i, prime(n)), where Kronecker(a, p) is the Kronecker symbol (the Legendre symbol for all odd primes) and H_i is the ith Heegner number (A003173 multiplied by 1).


EXAMPLE

We see that 3 = (1  sqrt(2))(1 + sqrt(2)) = (1/2  sqrt(11)/2)(1/2 + sqrt(11)/2) but is prime in Z[i], O_(Q(sqrt(7))), O_(Q(sqrt(19))), O_(Q(sqrt(43))), O_(Q(sqrt(67))) and O_(Q(sqrt(163))).
The fact that 3 = sqrt(3)^2 does not matter for our purpose here. So 3 splits in two of these domains but is prime in six of them.
As 3 is the second prime, a(2) is therefore 6 + 2 = 4.


MATHEMATICA

Table[Plus@@KroneckerSymbol[{1, 2, 3, 7, 11, 19, 43, 67, 163}, Prime[n]], {n, 100}]


CROSSREFS

Cf. A003173.
Sequence in context: A305403 A020805 A200587 * A106147 A202393 A073321
Adjacent sequences: A307833 A307834 A307835 * A307837 A307838 A307839


KEYWORD

sign


AUTHOR

Alonso del Arte, Jul 07 2019


STATUS

approved



