A002143 Class numbers h(-p) where p runs through the primes p == 3 (mod 4). (Formerly M2266 N0896)

1, 1, 1, 1, 3, 3, 1, 5, 3, 1, 7, 5, 3, 5, 3, 5, 5, 3, 7, 1, 11, 5, 13, 9, 3, 7, 5, 15, 7, 13, 11, 3, 3, 19, 3, 5, 19, 9, 3, 17, 9, 21, 15, 5, 7, 7, 25, 7, 9, 3, 21, 5, 3, 9, 5, 7, 25, 13, 5, 13, 3, 23, 11, 5, 5, 31, 13, 5, 21, 15, 5, 7, 9, 7, 33, 7, 21, 3, 29, 3, 31, 19, 5, 11, 15, 27, 17, 13

Offset

1,5

Comments

a(n) = h(-A002145(n)).

Same as (1/p)*(sum of quadratic nonresidues mod p in (0,p) - sum of quadratic residues mod p in (0,p)), for prime p == 3 (mod 4) if p > 3. (See Davenport's book and the first Mathematica program.) - Jonathan Sondow, Oct 27 2011

Conjecture: For any prime p > 3 with p == 3 (mod 8), we have 2*h(-p)*sqrt(p) = Sum_{k=1..(p-1)/2} csc(2*Pi*k^2/p). - Zhi-Wei Sun, Aug 06 2019

References

H. Davenport, Multiplicative Number Theory, Graduate Texts in Math. 74, 2nd ed., Springer, 1980, p. 51.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

T. D. Noe, Table of n, a(n) for n=1..10000

Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, vol. 101, (2002), pp. 105-114.

E. T. Ordman, Tables of the class number for negative prime discriminants, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes]

E. T. Ordman, Tables of the class number for negative prime discriminants, Math. Comp., 23 (1969), 458.

A. Pacetti and F. Rodriguez Villegas, Computing weight two modular forms of level p^2, Math. Comp. 74 (2004), 1545-1557. See Table 1.

N. Snyder, Lectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms. [Background information on class numbers, link sent by V. S. Miller, Nov 22 2009]

Wikipedia, Class numbers of quadratic fields

Formula

h(-p) = 1 + 2*sum(0 <= n <= (1/2)*sqrt(p/3)-1, d(n^2+n+(p+1)/4, [2*n+1, sqrt(n^2+n+(p+1)/4)])) for prime p=3 mod 4, p>3. d(n, [a, b])=card{d: d|n and a<d<b} for integer n and real a, b. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 19 2002

h(-p) = -(1/p)*sum(n=1..p-1, n*(n|p)) if p > 3, where (n|p) = +/- 1 is the Legendre symbol. - Jonathan Sondow, Oct 27 2011

h(-p) = (1/3)*sum(n=1..(p-1)/2, (n|p)) or sum(n=1..(p-1)/2, (n|p)) according as p == 3 or 7 (mod 8). - Jonathan Sondow, Feb 27 2012

Example

E.g., a(4) = 1 is the class number of -19, the 4th prime == 3 mod 4.
a(5) = -(1/23)*sum(n=1..22, n*(n|23)) = -(1/23)*(1 + 2 + 3 + 4 - 5 + 6 - 7 + 8 + 9 - 10 - 11 + 12 + 13 - 14 - 15 + 16 - 17 + 18 - 19 - 20 - 21 - 22) = 69/23 = 3. - Jonathan Sondow, Oct 27 2011

Mathematica

Cases[ Table[ With[ {p = Prime[n]}, If[ Mod[p, 4] == 3, -(1/p)*Sum[ a*JacobiSymbol[a, p], {a, 1, p - 1}]]], {n, 1, 100}], _Integer] (* Jonathan Sondow, Oct 27 2011 *)
p = Prime[n]; If[Mod[p, 4] == 3, NumberFieldClassNumber[Sqrt[-p]]] (* Jonathan Sondow, Feb 24 2012 *)

Prog

(PARI) forprime(p=3, 1e3, if(p%4==3, print1(qfbclassno(-p)", "))) \\ Charles R Greathouse IV, May 08 2011

Crossrefs

Cf. A002145 (primes p), A002146, A101435.

Sequence in context: A092674 A316366 A111945 * A039739 A160496 A105663

Adjacent sequences: A002140 A002141 A002142 * A002144 A002145 A002146

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 19 2002

Status

approved