cp's OEIS Frontend

Could also be called Gauss numbers, since he discovered them. Heegner proved list is complete. - Artur Jasinski, Mar 21 2003

Numbers n such that Q(sqrt(-n)) has unique factorization into primes.

These are the squarefree values of n for which if some positive integer N can be written in the form (a/2)^2+n*(b/2)^2 for integers a and b, then every prime factor P of N which occurs to an odd power can also be written in the form (c/2)^2+n*(d/2)^2 for integers c and d. - V. Raman, Sep 17 2012, May 01 2013

Cases n = 1 and n = 2 correspond to the rings Z[i] (Gaussian integers) and Z[sqrt(-2)] = numbers of the form a + b*sqrt(-2), where a and b are integers. Other cases, satisfying a(n) == 3 (mod 4), correspond to the rings of numbers of the form (a/2) + (b/2)*sqrt(-a(n)), for integers a and b of the same parity. All these rings admit unique factorization. - V. Raman, Sep 17 2012, corrected by Eric M. Schmidt, Feb 17 2013

The Heegner numbers greater than 3 can also be found using the Kronecker symbol, as follows: A number k > 3 is a Heegner number if and only if s = Sum_{j = 1..k} j * (j|k) is prime, which happens to be negative, where (x|y) is the Kronecker symbol. Also note for these results s = -k. But if s = -k is used as the selection condition (instead of primality), then the cubes of {7, 11, 19, 43, 67, 163} are also selected, followed by these same numbers to 9th power (and presumably followed by the 27th or 81st power). - Richard R. Forberg, Jul 18 2016

Theorem: The ring of integers of the imaginary quadratic field Q(sqrt(-n)) is Euclidean iff n = 1, 2, 3, 7 and 11. (Otherwise, the ring of integers of the imaginary quadratic field Q(sqrt(-n)) is principal iff n is a term of this sequence) [Link Stark-Heegner theorem]. - Bernard Schott, Feb 07 2020

Named after the German high school teacher and radio engineer Kurt Heegner (1893-1965). - Amiram Eldar, Jun 15 2021

These integers yield norm-Euclidean real quadratic fields. There are other positive integers, e.g., D=14 or D=69, for which Q[sqrt(D)] is Euclidean, but for a Euclidean function different from the field norm.

For further references see sequence A048981 which also lists negative D corresponding to (complex) norm-Euclidean fields. - M. F. Hasler, Jan 26 2014

These are norm-Euclidean fields, excluding for instance Q[sqrt(69)] which is Euclidean but not for norm. - Marc A. A. van Leeuwen, Feb 15 2011