login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002143 Class numbers h(-p) where p runs through the primes p == 3 (mod 4).
(Formerly M2266 N0896)
14

%I M2266 N0896 #57 Aug 06 2019 05:02:26

%S 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,

%T 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,

%U 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

%N Class numbers h(-p) where p runs through the primes p == 3 (mod 4).

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

%C 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

%C 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

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

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

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

%H T. D. Noe, <a href="/A002143/b002143.txt">Table of n, a(n) for n=1..10000</a>

%H Christian Aebi, Grant Cairns, <a href="http://arxiv.org/abs/1512.00896">Sums of Quadratic residues and nonresidues</a>, arXiv:1512.00896 [math.NT], 2015.

%H Kevin A. Broughan, <a href="http://www.math.waikato.ac.nz/~kab/papers/div4.pdf">Restricted divisor sums</a>, Acta Arithmetica, vol. 101, (2002), pp. 105-114.

%H E. T. Ordman, <a href="/A002143/a002143.pdf">Tables of the class number for negative prime discriminants</a>, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes]

%H E. T. Ordman, <a href="http://dx.doi.org/10.1090/S0025-5718-69-99644-6">Tables of the class number for negative prime discriminants</a>, Math. Comp., 23 (1969), 458.

%H A. Pacetti and F. Rodriguez Villegas, <a href="https://doi.org/10.1090/S0025-5718-04-01709-0">Computing weight two modular forms of level p^2</a>, Math. Comp. 74 (2004), 1545-1557. See Table 1.

%H N. Snyder, <a href="http://math.berkeley.edu/~nsnyder/tutorial/lecture7.pdf">Lectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms.</a> [Background information on class numbers, link sent by V. S. Miller, Nov 22 2009]

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Class_number_(number_theory)#Class_numbers_of_quadratic_fields">Class numbers of quadratic fields</a>

%F 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

%F 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

%F 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

%e E.g., a(4) = 1 is the class number of -19, the 4th prime == 3 mod 4.

%e 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

%t 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 *)

%t p = Prime[n]; If[Mod[p, 4] == 3, NumberFieldClassNumber[Sqrt[-p]]] (* _Jonathan Sondow_, Feb 24 2012 *)

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

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

%K nonn

%O 1,5

%A _N. J. A. Sloane_

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

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)