A002143 Class numbers h(-p) where p runs through the primes p == 3 (mod 4).
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
Keywords
Examples
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
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- E. T. Ordman, Tables of the class number for negative prime discriminants, Deposited in Unpublished Mathematical Table file of Math.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
Programs
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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 *) -
PARI
forprime(p=3, 1e3, if(p%4==3, print1(qfbclassno(-p)", "))) \\ Charles R Greathouse IV, May 08 2011
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
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
Extensions
More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 19 2002
Comments