A002324 - OEIS

%I M0016 N0002 #146 Sep 21 2024 10:16:02

%S 1,0,1,1,0,0,2,0,1,0,0,1,2,0,0,1,0,0,2,0,2,0,0,0,1,0,1,2,0,0,2,0,0,0,

%T 0,1,2,0,2,0,0,0,2,0,0,0,0,1,3,0,0,2,0,0,0,0,2,0,0,0,2,0,2,1,0,0,2,0,

%U 0,0,0,0,2,0,1,2,0,0,2,0,1,0,0,2,0,0,0,0,0,0,4,0,2,0,0,0,2,0,0,1,0,0,2,0,0,0,0,1,2,0,2,2,0,0

%N Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

%C 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

%C Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)*p^(-s) + Kronecker(m,p) * p^(-2s))^(-1) for m = -3.

%C (Number of points of norm n in hexagonal lattice) / 6, n>0.

%C 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.

%C 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

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 112, first display.

%D 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 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002324/b002324.txt">Table of n, a(n) for n = 1..10000</a>

%H G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.

%H Hershel M. Farkas, <a href="https://doi.org/10.1007/s11139-004-0141-5">On an arithmetical function</a>, Ramanujan J., 8(3) (2004), 309-315.

%H Pavel Guerzhoy and Ka Lun Wong, <a href="https://arxiv.org/abs/1905.06506">Farkas' identities with quartic characters</a>, arXiv:1905.06506 [math.NT], 2019.

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, 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.]

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016.

%H Gabriele Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>.

%H José Manuel Rodríguez Caballero, <a href="https://arxiv.org/abs/1709.09621">Divisors on overlapped intervals and multiplicative functions</a>, arXiv:1709.09621 [math.NT], 2017.

%H J. S. Rutherford, <a href="http://dx.doi.org/10.1007/BF01164477">Generating functions for the cage isomers of the C_{20n} icosahedral fullerenes, J. Mathematical Chem.</a>, 14 (1993), 385-390. [From _N. J. A. Sloane_, Mar 12 2009]

%H John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009

%H N. J. A. Sloane, <a href="/A002324/a002324.txt">Maple code for Dedekind zeta functions of quadratic number fields</a>.

%F From _N. J. A. Sloane_, Mar 22 2022 (Start):

%F The Dedekind zeta function DZ_K(s) for a quadratic field K of discriminant D is as follows.

%F Here m is defined by K = Q(sqrt(m)) (so m=D/4 if D is a multiple of 4, otherwise m=D).

%F DZ_K(s) is the product of three terms:

%F (a) Product_{odd primes p | D} 1/(1-1/p^s)

%F (b) Product_{odd primes p such that (D|p) = -1} 1/(1-1/p^(2s))

%F (c) Product_{odd primes p such that (D|p) = 1} 1/(1-1/p^s)^2

%F and if m is

%F    0,1,2,3,4,5,6,7 mod 8, the prime 2 is to be included in term

%F    -,c,a,a,-,b,a,a, respectively.

%F For Maple (and PARI) implementations, see link. (End)

%F 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

%F Has a nice Dirichlet series expansion, see PARI line.

%F G.f.: Sum_{k>0} x^k/(1+x^k+x^(2*k)). - _Vladeta Jovovic_, Dec 16 2002

%F a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = A033687(n). - _Michael Somos_, Apr 04 2003

%F 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

%F 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

%F 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

%F 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

%F a(n) = A001817(n) - A001822(n). - _R. J. Mathar_, Mar 31 2011

%F A004016(n) = 6*a(n) unless n=0.

%F 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

%F From _Andrey Zabolotskiy_, May 07 2018: (Start)

%F a(n) = Sum_{ m: m^2|n } A000086(n/m^2).

%F a(A003136(m)) > 0, a(A034020(m)) = 0 for all m. (End)

%F 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

%e 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 + ...

%p A002324 := proc(n)

%p     local a,pe,p,e;

%p     a :=1 ;

%p     for pe in ifactors(n)[2] do

%p         p := op(1,pe) ;

%p         e := op(2,pe) ;

%p         if p = 3 then

%p             ;

%p         elif modp(p,3) = 1 then

%p             a := a*(e+1) ;

%p         else

%p             a := a*(1+(-1)^e)/2 ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A002324(n),n=1..100) ; # _R. J. Mathar_, Sep 21 2024

%t 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 *)

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -3, #] &]]; (* _Michael Somos_, Aug 24 2014 *)

%t Table[DirichletConvolve[DirichletCharacter[3,2,m],1,m,n],{n,1,30}] (* _Steven Foster Clark_, May 29 2019 *)

%t 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 *)

%o (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_

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)))};

%o (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

%o (PARI) {a(n) = if( n<1, 0, qfrep([2,1; 1,2], n, 1)[n] / 3)}; \\ _Michael Somos_, Jun 05 2005

%o (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

%o (PARI) my(B=bnfinit(x^2+x+1)); vector(100,n,#bnfisintnorm(B,n)) \\ _Joerg Arndt_, Jun 01 2024

%o (Haskell)

%o a002324 n = a001817 n - a001822 n  -- _Reinhard Zumkeller_, Nov 26 2011

%o (Python)

%o from math import prod

%o from sympy import factorint

%o 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

%Y 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.

%Y Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

%Y Cf. A004016, A035019, A073010, A145377, A293899, A000086, A003136, A034020, A145394.

%K easy,nonn,nice,mult

%O 1,7

%A _N. J. A. Sloane_

%E More terms from _David Radcliffe_

%E Somos D.g.f. replaced with correct version by _Ralf Stephan_, Mar 27 2015