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A002143
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Class numbers h(-p) where p runs through the primes p == 3 (mod 4).
(Formerly M2266 N0896)
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14
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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
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 19 2002
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STATUS
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approved
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