login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS Foundation is grateful to everyone who made a donation during our Annual Appeal.     Visit the new and spectacular Pictures from the OEIS page!

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003173 Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).
(Formerly M0827)
34
1, 2, 3, 7, 11, 19, 43, 67, 163 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

For n = 1 and n = 2, the rings Z[i] (Gaussian Integers), and Z(sqrt(-2)) = numbers of the form a + b*sqrt(-2), where a and b are integers, admit unique factorization. - V. Raman, Sep 17 2012

For the values of n congruent to 3 (mod 4), the set of numbers of the form (a/2) + (b/2)*sqrt(-n), for integers a and b of the same parity, admit unique factorization. - V. Raman, Sep 17 2012, corrected by Eric M. Schmidt, Feb 17 2013

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 224.

N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103). See page 93.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 213.

Heegner K., 1952. Diophantische Analysis und Modulfunktionen. Matematische Zeitschrift Vol. 56 p. 227-253.

W. W. J. Hulsbergen, Conjectures in Arithmetic Algebraic Geometry, Vieweg, 1994, p. 8.

J. M. Masley, Where are the number fields with small class number?, pp. 221-242 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 295.

LINKS

Table of n, a(n) for n=1..9.

Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Eric Weisstein's World of Mathematics, Gauss's Class Number Problem [Yes, 3 s's in that URL]

Eric Weisstein's World of Mathematics, Heegner Number

Wikipedia, Heegner number

Index entries for sequences related to quadratic fields

FORMULA

a(n) = A263465(n) = -A048981(6-n) for n = 1, 2, 3, 4, 5. - Jonathan Sondow, Dec 09 2015

From Alexander R. Povolotsky, Dec 26 2015: (Start)

a(n) = 1+((1 + sqrt(3))^(n-1) - (1 - sqrt(3))^(n-1))/(2*sqrt(3)) for n = 1,2,3,4.

a(n) = 19+24*((1 + sqrt(3))^(n-6) - (1 - sqrt(3))^(n-6))/(2*sqrt(3)) for n = 6,7,8,9; so four almost integers (including one made famous by Ramanujan) could be expressed as: exp(Pi*sqrt(19+24*((1 + sqrt(3))^(n-6) - (1 - sqrt(3))^(n-6))/(2*sqrt(3)))) for n = 6,7,8,9.

In general a(n) = a(k) + (a(k+1)-a(k))*((1 + sqrt(3))^(n-k) - (1 - sqrt(3))^(n-k))/(2*sqrt(3)) where k=1 for n = 1,2,3,4 k=1 and k=6 for n = 6,7,8,9. (End)

MATHEMATICA

Union[ Select[ -NumberFieldDiscriminant[ Sqrt[-#]]& /@ Range[200], NumberFieldClassNumber[ Sqrt[-#]] == 1 & ] /. {4 -> 1, 8 -> 2}] (* Jean-Fran├žois Alcover, Jan 04 2012 *)

PROG

(PARI) select(n->qfbclassno(-n*if(n%4==3, 1, 4))==1, vector(200, i, i)) \\ Charles R Greathouse IV, Nov 20 2012

CROSSREFS

Cf. A014602 (for discriminants of these fields), A005847 (for class number 2), A048981, A003174, A263465.

Partitions into Heegner numbers: A242216, A242217.

Sequence in context: A158709 A180422 A055502 * A159262 A160434 A139630

Adjacent sequences:  A003170 A003171 A003172 * A003174 A003175 A003176

KEYWORD

fini,nonn,full,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 11 16:12 EST 2016. Contains 268184 sequences.