A003657 -id:A003657 - cp's OEIS frontend

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97, 101, 104, 105, 109, 113, 120, 124, 129, 133, 136, 137, 140, 141, 145, 149, 152, 156, 157, 161, 165, 168, 172, 173, 177, 181, 184, 185, 188, 193, 197

Offset: 1

N. J. A. Sloane, Mira Bernstein, Eric W. Weisstein

All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - Paul Muljadi, Mar 28 2008

Record numbers of prime divisors (with multiplicity) are 1, 5, and 4*A002110(n) for n > 0. - Charles R Greathouse IV, Jan 21 2022

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.
M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, page 432.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3001 from T. D. Noe)
Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Britta Habdank-Eichelsbacher, Unimodulare Gitter über Reell-Quadratischen Zahlkörpern, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Fundamental Discriminant.
Eric Weisstein's World of Mathematics, Class Number.

Cf. A003652, A003657, A002144, A003646 (class numbers), A014000, A014046, A086669, A232931, A290098.

Union of A039955 and 4*A230375.

fundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[SquareFreeQ[d] && d != 1]]; m = d/4; Return[SquareFreeQ[m] && Mod[m, 4] > 1]; ]; Join[{1}, Select[Range[200], fundamentalDiscriminantQ]] (* Jean-François Alcover, Nov 02 2011, after Eric W. Weisstein *)
Select[Range[200], NumberFieldDiscriminant@Sqrt[#] == # &]  (* Alonso del Arte, Apr 02 2014, based on Arkadiusz Wesolowski's program for A094612 *)
max = 200; Drop[Select[Union[Table[Abs[MoebiusMu[n]] * n * 4^Boole[Not[Mod[n, 4] == 1]], {n, max}]], # < max &], 1] (* Alonso del Arte, Apr 02 2014 *)

v=[]; for(n=1,500,if(isfundamental(n),v=concat(v,n))); v

list(lim)=my(v=List()); forsquarefree(n=1,lim\4, listput(v, if(n[1]%4==1, n[1], 4*n[1]))); forsquarefree(n=lim\4+1, lim\1, if(n[1]%4==1, listput(v,n[1]))); Set(v) \\ Charles R Greathouse IV, Jan 21 2022

def is_fundamental(d):
    r = d % 4
    if r > 1 : return False
    if r == 1: return (d != 1) and is_squarefree(d)
    q = d // 4
    return is_squarefree(q) and (q % 4 > 1)
[1] + [n for n in (1..200) if is_fundamental(n)] # Peter Luschny, Oct 15 2018

Squarefree numbers (multiplied by 4 if not == 1 (mod 4)).

a(n) ~ (Pi^2/3)*n. There are (3/Pi^2)*x + O(sqrt(x)) terms up to x. - Charles R Greathouse IV, Jan 21 2022

More terms from Eric W. Weisstein and Jason Earls, Jun 19 2001

A039957 Squarefree numbers congruent to 3 mod 4.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 31, 35, 39, 43, 47, 51, 55, 59, 67, 71, 79, 83, 87, 91, 95, 103, 107, 111, 115, 119, 123, 127, 131, 139, 143, 151, 155, 159, 163, 167, 179, 183, 187, 191, 195, 199, 203, 211, 215, 219, 223, 227, 231, 235, 239, 247, 251, 255

Offset: 1

R. K. Guy

Negatives of odd fundamental discriminants D := b^2-4*a*c<0 of definite integer binary quadratic forms F=a*x^2+b*x*y+c*y^2. See Buell reference pp. 224-230. See 4*A089269 = A191483 for the even case and A003657 for combined even and odd numbers. - Wolfdieter Lang, Nov 07 2003

The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la forme t^2+au^2 a étant un nombre de la forme 4n-1, Essai sur la Théorie des Nombres An VI, Table V.

Cf. A039955, A039956, A185197, A191483.

a039957 n = a039957_list !! (n-1)
a039957_list = filter ((== 3) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

[4*n+3: n in [0..63] | IsSquarefree(4*n+3)];  // Bruno Berselli, Mar 04 2011

fQ[n_] := SquareFreeQ[n] && Mod[n, 4] == 3; Select[ Range@ 258, fQ] (* Robert G. Wilson v, Mar 02 2011 *)
Select[Range[3,300,4],SquareFreeQ] (* Harvey P. Dale, Mar 08 2015 *)

is(n)=n%4==3 && issquarefree(n) \\ Charles R Greathouse IV, Feb 07 2017

Offset corrected

A114643 Number of real primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1

Offset: 1

Steven Finch, Feb 16 2006

a(n) = 1 if either n or -n is a fundamental discriminant (not both); a(n) = 2 if n and -n are fundamental discriminants; a(n) = 0 otherwise. Also, Sum_{k=1..n} a(k) is asymptotic to (6/Pi^2)*n.
From Jianing Song, Feb 27 2019: (Start)
If n is an odd squarefree number, then a(n) = 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(n,k)} = {Jacobi(k,n)} if n == 1 (mod 4) and {Kronecker(-n,k)} = {Jacobi(k,n)} if n == 3 (mod 4).
If n = 4*m, m is an odd squarefree number, then a(n) is also 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(-n,k)} if m == 1 (mod 4) and {Kronecker(n,k)} if m == 3 (mod 4).
If n is 8 times an odd squarefree number, then a(n) = 2, where the two real primitive Dirichlet characters modulo n are {Kronecker(n,k)} and {Kronecker(-n,k)}.
a(n) = 0 if n == 2 (mod 4), n is divisible by 16 or the square of an odd prime. (End)
Mobius transform of A060594. - Jianing Song, Mar 02 2019

			From _Jianing Song_, Feb 27 2019: (Start)
For n = 5, the only real primitive Dirichlet characters modulo n is {Kronecker(5,k)} = [0, 1, -1, -1, 1] = A080891, so a(5) = 1.
For n = 8, the real primitive Dirichlet characters modulo n are {Kronecker(8,k)} = [0, 1, 0, -1, 0, -1, 0, 1] = A091337 and [0, 1, 0, 1, 0, -1, 0, -1] = A188510, so a(8) = 2.
For n = 20, the only real primitive Dirichlet characters modulo n is {Kronecker(-20,k)} = [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1] = A289741, so a(20) = 1. (End)

W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, pp. 224-226.

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Cubic and quartic characters [Broken link]
Steven R. Finch, Cubic and quartic characters.
Vaclav Kotesovec, Graph - the asymptotic ratio
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
I. J. Zucker and M. M. Robertson, Some properties of Dirichlet L-series, J. Phys. A 9 (1976) 1207-1214.

Cf. A003657, A003658.
Cf. A160498 (number of cubic primitive Dirichlet characters modulo n), A160499 (number of quartic primitive Dirichlet characters modulo n).
Cf. A060594 (number of solutions to x^2 == 1 (mod n)).

A114643 := proc(n)
    local a,pf,p,r;
    a := 1 ;
    for pf in ifactors(n)[2] do
        p := op(1,pf);
        r := op(2,pf);
        if p = 2 then
            if r =  1 then
                a := 0 ;
            elif r =  2 then
                ;
            elif r =  3 then
                a := a*2 ;
            elif r >=  4 then
                a := 0 ;
            end if;
        else
            if r =1 then
                ;
            else
                a := 0 ;
            end if;
        end if;
    end do:
    a ;
end proc:
seq(A114643(n),n=1..40) ; # R. J. Mathar, Mar 02 2015
# Alternative:
f:= proc(n) local r,v,F;
  v:= padic:-ordp(n,2);
  if v = 1 or v >= 4 then return 0
  elif v = 3 then r:= 2
  else r:= 1
  fi;
  if numtheory:-issqrfree(n/2^v) then r else 0 fi
end proc:
map(f, [$1..100]); # Robert Israel, Oct 08 2017

a[n_] := Sum[ MoebiusMu[n/d] * Sum[ If[ Mod[i^2 - 1, d] == 0, 1, 0], {i, 2, d}], {d, Divisors[n]}]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 20 2013, after Steven Finch *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e >= 4, 0]; f[p_, e_] := If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^2-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

This sequence is multiplicative with a(2) = 0, a(4) = 1, a(8) = 2, a(2^r) = 0 for r > 3, a(p) = 1 for prime p > 2 and a(p^r) = 0 for r > 1. - Steven Finch, Mar 08 2006 (With correction by Jianing Song, Jun 28 2018)

Dirichlet g.f.: zeta(s)*(1 + 2^(-2s) + 2^(1-3s))/(zeta(2s)*(1 + 2^(-s))). - R. J. Mathar, Jul 03 2011

A089269 Squarefree numbers congruent to 1 or 2 mod 4.

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 17, 21, 22, 26, 29, 30, 33, 34, 37, 38, 41, 42, 46, 53, 57, 58, 61, 62, 65, 66, 69, 70, 73, 74, 77, 78, 82, 85, 86, 89, 93, 94, 97, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, 130, 133, 134, 137, 138, 141, 142, 145, 146, 149, 154, 157

Offset: 1

Wolfdieter Lang, Nov 07 2003

a(n) = one-fourth of the (negated) fundamental even discriminants D := b^2-4*a*c<0 of positive definite binary quadratic forms F=a*x^2+b*x*y+c*y^2. See A039957 for the odd numbers and A003657 for the combined even and odd numbers.

The asymptotic density of this sequence is 4/Pi^2 (A185199). - Amiram Eldar, Feb 23 2021

Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989, pp. 231-234.
Arnold Scholz and Bruno Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, Ch. 30.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A000003, A005117, A039957, A003657, A185199.

[n: n  in [1..200] | IsSquarefree(n) and n mod 4 in [1,2]]; // Vincenzo Librandi, Oct 20 2017

Select[Range[200], MemberQ[{1, 2}, Mod[#, 4]]&& SquareFreeQ[#]&] (* Vincenzo Librandi, Oct 20 2017 *)

Entry revised by N. J. A. Sloane, May 28 2014

A191483 Even discriminants of imaginary quadratic fields, negated.

Original entry on oeis.org

4, 8, 20, 24, 40, 52, 56, 68, 84, 88, 104, 116, 120, 132, 136, 148, 152, 164, 168, 184, 212, 228, 232, 244, 248, 260, 264, 276, 280, 292, 296, 308, 312, 328, 340, 344, 356, 372, 376, 388, 404, 408, 420, 424, 436, 440, 452, 456, 472, 488, 516, 520, 532, 536, 548

Offset: 1

Robert G. Wilson v, Jun 04 2011

Terms of A003657 that are not in A039957.

The asymptotic density of this sequence is 1/Pi^2 (A092742). - Amiram Eldar, Feb 23 2021

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A003657, A039957, A089269, A092742.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *); -Select[-Range@550, FundamentalDiscriminantQ@# && EvenQ@# &]
(* Second program: *)
Select[Range[600], Mod[#, 4] == 0 && SquareFreeQ[#/4] && Mod[#, 16] != 12&] (* Jean-François Alcover, Jul 25 2019, after Andrew Howroyd *)

ok(n)={isfundamental(-n) && n%2==0} \\ Andrew Howroyd, Jul 25 2018

ok(n)={n%4==0 && issquarefree(n/4) && n%16<>12} \\ Andrew Howroyd, Jul 25 2018

Complement(A003657, A039957).

a(n) = 4*A089269(n). - Andrew Howroyd, Jul 25 2018

A354257 a(n) is the smallest k such that there exists a degree-k primitive Dirichlet characters modulo n, or -1 no such k exists.

Original entry on oeis.org

1, -1, 2, 2, 2, -1, 2, 2, 3, -1, 2, 2, 2, -1, 2, 4, 2, -1, 2, 2, 2, -1, 2, 2, 5, -1, 9, 2, 2, -1, 2, 8, 2, -1, 2, 6, 2, -1, 2, 2, 2, -1, 2, 2, 6, -1, 2, 4, 7, -1, 2, 2, 2, -1, 2, 2, 2, -1, 2, 2, 2, -1, 3, 16, 2, -1, 2, 2, 2, -1, 2, 6, 2, -1, 10, 2, 2, -1, 2, 4, 27, -1, 2, 2, 2, -1, 2, 2, 2, -1

Offset: 1

Jianing Song, May 21 2022

sign

For n !== 2 (mod 4), a(n) is the smallest k such that A354058(n,k) != 0 (or the smallest k such that A354061(n,k) != 0).

For n !== 2 (mod 4), a(n) is the smallest k such that Sum_{d|n} mu(n/d)*#{x in (Z/dZ)*: x^k == 1 (mod d)} != 0, where mu = A008683, (Z/dZ)* is the multiplicative group of integers modulo d.

			a(45) = 6: there does not exist a linear, quadratic, cubic, quartic or quintic primitive Dirichlet characters modulo 45, but there are 4 sextic primitive Dirichlet characters.
a(63) = 3: there does not exist a linear or quadratic primitive Dirichlet characters modulo 63, but there are 4 cubic primitive Dirichlet characters.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A354058, A354061, A008683.
A354258 gives the earliest occurrence of each positive integers.
Indices of 2: A003657 U A003658 \ {1}.

a(n) = if(n%4==2, return(-1), my(e_0 = valuation(n,2)); n=n>>e_0; my(L=factor(n),w=omega(n),v=[],M=1); for(j=1, w, if(L[j,2]==1, v=concat(v,j), M*=L[j,1]^(L[j,2]-1))); if(e_0 >= 2, return(2^max(e_0-2,1)*M), for(i=1, #v, if(gcd(M,L[v[i],1]-1)==1, return(2*M))); return(M)))

Write n = 2^(e_0) * (p_1) * ... * (p_r) * (q_1)^(e_1) * ... * (q_s)^(e_s), where (p_i)'s and (q_j)'s are distinct odd primes, e_j >= 2. Let M = Product_{j=1..s} (q_j)^(e_j-1):
(i) if e_0 = 1, then a(n) = -1;
(ii) if e_0 = 2, then a(n) = 2*M;
(iii) if e_0 >= 3, then a(n) = 2^(e_0-2)*M;
(iv) if e_0 = 0, then a(n) = M if for every 1 <= i <= r, there exists 1 <= j <= s such that q_j divides p_i - 1; otherwise a(n) = 2*M.
Let k >= 1. Write k = 2^(e_0) * (q_1)^(e_1) * ... * (q_s)^(e_s), e_j >= 1. Let N = Product_{j=1..s} (q_j)^(e_j+1):
(i) if e_0 = 0, then a(n) = k <=> n = (p_1) * ... * (p_r) * N, where: p_i != q_j, and for every 1 <= i <= r, there exists 1 <= j <= s such that q_j divides p_i - 1.
(ii) if e_0 = 1, then a(n) = k <=> (a) n = (p_1) * ... * (p_r) * N, where: p_i != q_j, and there exists 1 <= i <= r such that none of (q_j)'s divides p_i - 1; or (b) n = (4 or 8) * N * (an odd squarefree number coprime to N);
(iii) if e_0 >= 2, then a(n) = k <=> n = 2^(e_0+2) * N * (an odd squarefree number coprime to N).

A191408 Duplicate of A006641.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 2, 4, 2, 1, 5, 2, 2, 4, 4, 3, 1, 4, 7, 5, 3, 4, 6, 2, 2, 8, 5, 6, 3, 8, 2, 6, 10, 4, 2, 5, 5, 4, 4, 3, 10, 2, 7, 6, 4, 10, 1, 8, 11, 4, 5, 8, 4, 2, 13, 4, 9, 4, 3, 6, 14, 4, 7, 5, 4, 12, 2, 2, 15, 6, 6, 8, 7, 12, 4, 8, 13, 8, 2, 11, 8, 4, 3, 14, 4, 4, 8, 10, 8

Offset: 1

Robert G. Wilson v, Jun 01 2011

dead

Same as A006641. - Georg Fischer, Oct 12 2018

Cf. A191410.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric W. Weisstein *);
NumberFieldClassNumber@ Sqrt@ # & /@ Select[-Range@ 300, FundamentalDiscriminantQ]

for(n=1, 300, if(isfundamental(-n), print1(quadclassunit(-n).no, ", "))) \\ Andrew Howroyd, Jul 23 2018

Class number of A003657(n).

Terms corrected by Andrew Howroyd and Robert G. Wilson v, Jul 24 2018

A306503 a(n) is the index in primes of A306500(n), or 0 if A306500(n) = 0.

Original entry on oeis.org

23338590792, 2946, 1, 2, 3, 1, 5, 2, 1, 3, 1, 5, 1, 5, 16, 1, 6, 7, 1, 2, 3, 38, 2, 1, 1, 5, 3, 1, 11, 11, 1, 1, 2, 9, 1, 13, 2, 1, 6, 15, 1, 3, 8, 4, 5, 1, 43, 1, 2, 8, 1, 1760, 2, 1, 10, 3, 1, 13, 29, 1, 5, 1, 11, 17, 2, 1, 8, 1, 5, 16, 1, 171, 33, 1, 4, 1, 2, 3

Offset: 1

Jianing Song, Feb 19 2019

nonn

a(n) is the smallest integer k such that Sum_{i=1..k} Kronecker(-A003657(n),prime(i)) > 0, or 0 if no such k exists.

See A306500 for the actual primes.

			See A306500 for the example that shows a(18) = 7.

Cf. A003657, A306500, A306502.

b(n) = my(i=0); forprime(p=2, oo, i+=kronecker(n, p); if(i>0, return(p)))
print1(23338590792, ", "); for(n=4, 300, if(isfundamental(-n), print1(primepi(b(-n)), ", ")))

A329648 Let D = A014601(n) be the n-th positive integer congruent to 0 or 3 mod 4, then a(n) = b(D) := -Sum_{i=1..D} Kronecker(-D,i)*i, where Kronecker(-D,i) is the Kronecker symbol.

Original entry on oeis.org

1, 2, 7, 8, 11, 8, 30, 8, 19, 40, 69, 48, 9, 0, 93, 32, 70, 36, 156, 80, 43, 88, 235, 32, 102, 104, 220, 224, 177, 0, 126, 32, 67, 272, 497, 0, 50, 152, 395, 160, 249, 336, 522, 176, 182, 0, 760, 192, 0, 0, 515, 624, 321, 72, 888, 0, 230, 696, 1190, 480, 246, 0, 635

Offset: 1

Jianing Song, Nov 18 2019

nonn

Note that {Kronecker(D,i)} is a Dirichlet character mod |D| if and only if D == 0, 1 (mod 4).

We have the identity: -Sum_{i=1..D} Kronecker(-D,i)*i^2 = D*b(D). Proof: -Sum_{i=1..D} Kronecker(-D,i)*i^2 = -(1/2)*Sum_{i=1..D} (Kronecker(-D,i)*i^2+Kronecker(-D,D-i)*(D-i)^2) = -(1/2)*Sum_{i=1..D} (Kronecker(-D,i)*(i^2-(D-i)^2)) = -(1/2)*Sum_{i=1..D} (Kronecker(-D,i)*(2*D*i-D^2) = D*b(D) + (D^2/2)*(Sum_{i=1..D} Kronecker(-D,i)) = D*b(D).

			For n = 7, D = 15, b(15) = -(1 + 2 + 4 - 7 + 8 - 11 - 13 - 14) = 30, which is equal to 15*h(-15). Note that the class number of Q[sqrt(-15)] is 2.
For D < 100, b(D) = 0 for D = 28 = 7*2^2, 60 = 15*2^2, 72 = 8*3^2, 92 = 23*2^2, 99 = 11*3^2 and 100 = 4*5^2, where -7, -15, -8, -23, -11 and -4 are fundamental discriminants. Note that Kronecker(-7,2) = Kronecker(-15,2) = Kronecker(-8,3) = Kronecker(-23,2) = Kronecker(-11,3) = 1. On the other hand, for D = 213444 = 4*231^2, we have c(213444) = 2*h(-4)/w(-4) * (1-Kronecker(-4,3))*(1-Kronecker(-4,7))*(1-Kronecker(-4,11)) = 4 and b(213444) = 213444*4 = 853776.

Jianing Song, Table of n, a(n) for n = 1..10000
Jianing Song, Notes on the value of -(1/D)*(Sum_{i=1..D} Kronecker(-D,i)*i)
Eric Weisstein's World of Mathematics, Class Number
Eric Weisstein's World of Mathematics, Dirichlet L-Series

Cf. A014601, A003657, A001221.

b[n_] = -Sum[KroneckerSymbol[n, i]*i, {i, 1, n}];
a[n_] = b[2 n + Mod[n, 2]]

b(n) = -sum(i=1, n, kronecker(-n,i)*i)
a(n) = b(2*n + (n%2))

Let c(D) = b(D)/D = -(1/D)*(Sum_{i=1..D} Kronecker(-D,i)*i). Let -d be the unique fundamental discriminant (i.e., d is in A003657) such that D/d is a square, then c(D) = 2*h(-d)/w(-d) * Product_{primes p|D} (1-Kronecker(-d,p)), where h(-d) is the class number of K = Q[sqrt(-d)], w(-d) is the number of elements in K whose norms are 1 (w(-d) = 6 if d = 3, 4 if d = 4 and 2 if d > 4). This can be seen as the generalization of the well known class number formula: if -d is a fundamental discriminant then c(d) = 2*h(-d)/w(-d). See my notes in the Links section.
b(D) = 0 if and only if there exists a prime p being a factor of D such that if we write D = p^e * s, gcd(p,s) = 1, then e is even and Kronecker(-s,p) = 1; if p = 2, then s == 7 (mod 8).
If -d is a fundamental discriminant, then Sum_{k>=1} Kronecker(-d,k)/k = 2*Pi*h(-d)/(sqrt(d)*w(-d)) = Pi*c(d)/sqrt(d) = Pi*b(d)/d^(3/2). Here Sum_{k>=1} Kronecker(-d,k)/k is the value of the Dirichlet L-series of a non-principal character modulo d at s=1.

A225365 Negative fundamental discriminants with non-isomorphic class groups (negated).

Original entry on oeis.org

3, 15, 23, 39, 47, 71, 84, 87, 95, 119, 167, 191, 199, 215, 231, 239, 260, 311, 327, 335, 383, 399, 407, 420, 431, 455, 479, 551, 591, 647, 671, 695, 719, 759, 776, 791, 831, 839, 887, 935, 959, 983, 1031, 1079, 1140, 1151, 1199, 1239, 1271, 1295, 1319, 1391, 1439, 1487, 1511, 1559, 1679, 1751, 1799, 1847, 1959, 1991, 2015

Offset: 1

Rick L. Shepherd, May 05 2013

nonn

More precisely, the least absolute values of negative fundamental discriminants with class groups non-isomorphic to all class groups for negative fundamental discriminants with smaller absolute values.

The n-th line of the linked file gives the invariant factor decomposition of the class group corresponding to the fundamental discriminant -a(n).

			The fundamental discriminant -3 corresponds to the trivial class group. The fundamental discriminant -15 is the first negative fundamental discriminant encountered (least absolute value) whose class group has a different structure, isomorphic to Z_2. The fundamental discriminant -84 is the one with least absolute value whose class group is isomorphic to the Klein-4 group.

Rick L. Shepherd and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 7194 terms from Shepherd)
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Rick L. Shepherd, Invariant factor decompositions for corresponding class groups

Cf. A003657.

{allocatemem(32000000);
\\ Increase precision to more than 100 digits to go beyond 7194 terms.
default(realprecision, 100);
terms_wanted = 7194;
G = Set(); k = 0;
while(length(G)1, print("Certify failed for ", -k, " -- exiting (", length(G), " terms found)"); break);
    if(setsearch(G, F.clgp.cyc)==0,
      G = setunion(G, [F.clgp.cyc]);
      write("b225365.txt", length(G), " ", k);
      write("a225365.txt", length(G), " ", F.clgp.cyc);
      if(length(G)%100==0, print1("...", length(G), "... ")))))}

cp's OEIS Frontend

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

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A039957 Squarefree numbers congruent to 3 mod 4.

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A114643 Number of real primitive Dirichlet characters modulo n.

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A089269 Squarefree numbers congruent to 1 or 2 mod 4.

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A191483 Even discriminants of imaginary quadratic fields, negated.

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A354257 a(n) is the smallest k such that there exists a degree-k primitive Dirichlet characters modulo n, or -1 no such k exists.

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A191408 Duplicate of A006641.