OFFSET
1,7
COMMENTS
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -3. See Formula section for the general expression. - N. J. A. Sloane, Mar 22 2022
Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)*p^(-s) + Kronecker(m,p) * p^(-2s))^(-1) for m = -3.
(Number of points of norm n in hexagonal lattice) / 6, n>0.
The hexagonal lattice is the familiar 2-dimensional lattice (A_2) in which each point has 6 neighbors. This is sometimes called the triangular lattice.
The first occurrence of a(n) = 1, 2, 3, 4,... is at n= 1, 7, 49, 91, 2401, 637, ... as tabulated in A343771. - R. J. Mathar, Sep 21 2024
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 112, first display.
J. W. L. Glaisher, Table of the excess of the number of (3k+1)-divisors of a number over the number of (3k+2)-divisors, Messenger Math., 31 (1901), 64-72.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
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
G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
Hershel M. Farkas, On an arithmetical function, Ramanujan J., 8(3) (2004), 309-315.
Pavel Guerzhoy and Ka Lun Wong, Farkas' identities with quartic characters, arXiv:1905.06506 [math.NT], 2019.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.
J. S. Rutherford, Generating functions for the cage isomers of the C_{20n} icosahedral fullerenes, J. Mathematical Chem., 14 (1993), 385-390. [From N. J. A. Sloane, Mar 12 2009]
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From N. J. A. Sloane, Feb 23 2009
N. J. A. Sloane, Maple code for Dedekind zeta functions of quadratic number fields.
FORMULA
From N. J. A. Sloane, Mar 22 2022 (Start):
The Dedekind zeta function DZ_K(s) for a quadratic field K of discriminant D is as follows.
Here m is defined by K = Q(sqrt(m)) (so m=D/4 if D is a multiple of 4, otherwise m=D).
DZ_K(s) is the product of three terms:
(a) Product_{odd primes p | D} 1/(1-1/p^s)
(b) Product_{odd primes p such that (D|p) = -1} 1/(1-1/p^(2s))
(c) Product_{odd primes p such that (D|p) = 1} 1/(1-1/p^s)^2
and if m is
0,1,2,3,4,5,6,7 mod 8, the prime 2 is to be included in term
-,c,a,a,-,b,a,a, respectively.
For Maple (and PARI) implementations, see link. (End)
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 3*v^2 + 4*w^2 - 2*u*w + w - v. - Michael Somos, Jul 20 2004
Has a nice Dirichlet series expansion, see PARI line.
G.f.: Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Vladeta Jovovic, Dec 16 2002
a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = A033687(n). - Michael Somos, Apr 04 2003
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - u3)*(u3 - u6) - (u2 - u6)^2. - Michael Somos, May 20 2005
Multiplicative with a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3). - Michael Somos, May 20 2005
G.f.: Sum_{k>0} x^(3*k - 2) / (1 - x^(3*k - 2)) - x^(3*k - 1) / (1 - x^(3*k - 1)). - Michael Somos, Nov 02 2005
G.f.: Sum_{n >= 1} q^(n^2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). - Jeremy Lovejoy, Jun 12 2009
A004016(n) = 6*a(n) unless n=0.
Dirichlet g.f.: zeta(s)*L(chi_2(3),s), with chi_2(3) the nontrivial Dirichlet character modulo 3 (A102283). - Ralf Stephan, Mar 27 2015
From Andrey Zabolotskiy, May 07 2018: (Start)
a(n) = Sum_{ m: m^2|n } A000086(n/m^2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 11 2022
EXAMPLE
G.f. = x + x^3 + x^4 + 2*x^7 + x^9 + x^12 + 2*x^13 + x^16 + 2*x^19 + 2*x^21 + ...
MAPLE
A002324 := proc(n)
local a, pe, p, e;
a :=1 ;
for pe in ifactors(n)[2] do
p := op(1, pe) ;
e := op(2, pe) ;
if p = 3 then
;
elif modp(p, 3) = 1 then
a := a*(e+1) ;
else
a := a*(1+(-1)^e)/2 ;
end if;
end do:
a ;
end proc:
seq(A002324(n), n=1..100) ; # R. J. Mathar, Sep 21 2024
MATHEMATICA
dn12[n_]:=Module[{dn=Divisors[n]}, Count[dn, _?(Mod[#, 3]==1&)]-Count[ dn, _?(Mod[#, 3]==2&)]]; dn12/@Range[120] (* Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Aug 24 2014 *)
Table[DirichletConvolve[DirichletCharacter[3, 2, m], 1, m, n], {n, 1, 30}] (* Steven Foster Clark, May 29 2019 *)
f[3, p_] := 1; f[p_, e_] := If[Mod[p, 3] == 1, e+1, (1+(-1)^e)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k + x^(2*k)), x * O(x^n)), n))}; \\ Michael Somos
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)))};
(PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==3, 1, if( p%3==1, e+1, !(e%2))))))}; \\ Michael Somos, May 20 2005
(PARI) {a(n) = if( n<1, 0, qfrep([2, 1; 1, 2], n, 1)[n] / 3)}; \\ Michael Somos, Jun 05 2005
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker(-3, p)*X))[n])}; \\ Michael Somos, Jun 05 2005
(PARI) my(B=bnfinit(x^2+x+1)); vector(100, n, #bnfisintnorm(B, n)) \\ Joerg Arndt, Jun 01 2024
(Haskell)
a002324 n = a001817 n - a001822 n -- Reinhard Zumkeller, Nov 26 2011
(Python)
from math import prod
from sympy import factorint
def A002324(n): return prod(e+1 if p%3==1 else int(not e&1) for p, e in factorint(n).items() if p != 3) # Chai Wah Wu, Nov 17 2022
CROSSREFS
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
KEYWORD
easy,nonn,nice,mult,changed
AUTHOR
EXTENSIONS
More terms from David Radcliffe
Somos D.g.f. replaced with correct version by Ralf Stephan, Mar 27 2015
STATUS
approved