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A002143
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Class numbers h(-p) where p runs though the primes p == 3 (mod 4).
(Formerly M2266 N0896)
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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| a(n) = h(-A002145(n)).
Same as -(1/p)*(sum of quadratic residues mod p in [1,p-1] - sum of nonresidues mod p in [1,p-1]), for prime p == 3 (mod 4) if p > 3. (See Davenport's book and the Mathematica program.) - Jonathan Sondow, Oct 27 2011
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REFERENCES
| H. Davenport, Multiplicative Number Theory, Graduate Texts in Math. 74, 2nd ed., Springer, 1980, p. 51.
E. T. Ordman, Tables of the class number for negative prime discriminants, Math. Comp., 23 (1969), 458.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, vol. 101, (2002), pp. 105-114.
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
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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
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EXAMPLE
| E.g. a(4) = 1 is the class number of -19, the 4-th 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
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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 *)
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PROG
| (PARI) forprime(p=3, 1e3, if(p%4==3, print1(qfbclassno(-p)", "))) \\ Charles R Greathouse IV, May 08, 2011
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CROSSREFS
| Cf. A002145 (primes p), A002146
Sequence in context: A111408 A092674 A111945 * A039739 A160496 A105663
Adjacent sequences: A002140 A002141 A002142 * A002144 A002145 A002146
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 19 2002
Editorial comments from M. F. Hasler, Nov 22 2009
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