

A002143


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


14



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

1,5


COMMENTS

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..(p1)/2} csc(2*Pi*k^2/p).  ZhiWei 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



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: dn 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..p1, n*(np)) if p > 3, where (np) = +/ 1 is the Legendre symbol.  Jonathan Sondow, Oct 27 2011
h(p) = (1/3)*sum(n=1..(p1)/2, (np)) or sum(n=1..(p1)/2, (np)) 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*(n23)) = (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



CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



