A145377 -id:A145377 - cp's OEIS frontend

A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

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, 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, 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

Offset: 1

N. J. A. Sloane

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

			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 + ...

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).

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.

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

a002324 n = a001817 n - a001822 n  -- Reinhard Zumkeller, Nov 26 2011

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

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 *)

{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

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

{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

{a(n) = if( n<1, 0, qfrep([2,1; 1,2], n, 1)[n] / 3)}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker(-3, p)*X))[n])}; \\ Michael Somos, Jun 05 2005

my(B=bnfinit(x^2+x+1)); vector(100,n,#bnfisintnorm(B,n)) \\ Joerg Arndt, Jun 01 2024

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

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
a(n) = A001817(n) - A001822(n). - R. J. Mathar, Mar 31 2011
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).
a(A003136(m)) > 0, a(A034020(m)) = 0 for all m. (End)
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

More terms from David Radcliffe

Somos D.g.f. replaced with correct version by Ralf Stephan, Mar 27 2015

A088534 Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Nov 16 2003

nonn

Also, apparently the number of 6-regular plane graphs with n vertices that have only trigonal faces and loops ("({1,3},6)-spheres" from the paper by Michel Deza and Mathieu Dutour Sikiric). - Andrey Zabolotskiy, Dec 22 2021

			From _M. F. Hasler_, Mar 05 2018: (Start)
a(0) = a(1) = 1 since 0 = 0^2 + 0*0 + 0^2 and 1 = 0^2 + 0*1 + 1^2.
a(2) = 0 since 2 cannot be written as x^2 + xy + y^2.
a(49) = 2 since 49 = 0^2 + 0*7 + 7^2 = 3^2 + 3*5 + 5^2. (End)

B. C. Berndt, "On a certain theta-function in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992), 33-43.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Michel Deza and Mathieu Dutour Sikiric, Zigzag and Central Circuit Structure of ({1,2,3},6)-spheres, Taiwanese Journal of Mathematics, 16 (June 2012), No. 3, 913-940; see Table 1, p. 916.
Michel-Marie Deza, Mathieu Dutour Sikiric, and Mikhail Ivanovitch Shtogrin, Geometric Structure of Chemistry-Relevant Graphs, Springer, 2015; see Table 5.4 and Section 5.4.
Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005.

Cf. A003136, A034020, A000196.
Cf. A118886 (indices of values > 1), A198772 (indices of 1's), A198773 (indices of 2's), A198774 (indices of 3's), A198775 (indices of 4's), A198799 (index of 1st term = n).
Cf. A215622.

a088534 n = length
   [(x,y) | y <- [0..a000196 n], x <- [0..y], x^2 + x*y + y^2 == n]
a088534_list = map a088534 [0..]
-- Reinhard Zumkeller, Oct 30 2011

function A088534(n)
    n % 3 == 2 && return 0
    M = Int(round(2*sqrt(n/3)))
    count = 0
    for y in 0:M, x in 0:y
        n == x^2 + y^2 + x*y && (count += 1)
    end
    return count
end
A088534list(upto) = [A088534(n) for n in 0:upto]
A088534list(104) |> println # Peter Luschny, Mar 17 2018

a[n_] := Sum[Boole[i^2 + i*j + j^2 == n], {i, 0, n}, {j, 0, i}];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jun 20 2018 *)

a(n)=sum(i=0,n,sum(j=0,i,if(i^2+i*j+j^2-n,0,1)))

A088534(n,d)=sum(x=0,sqrt(n\3),sum(y=max(x,sqrtint(n-x^2)\2),sqrtint(n-2*x^2),x^2+x*y+y^2==n&&(!d||!printf("%d",[x,y]))))\\ Set 2nd arg = 1 to print all decompositions, with 0 <= x <= y. - M. F. Hasler, Mar 05 2018

def A088534(n):
    c = 0
    for y in range(n+1):
        if y**2 > n:
            break
        for x in range(y+1):
            z = x*(x+y)+y**2
            if z > n:
                break
            elif z == n:
                c += 1
    return c # Chai Wah Wu, May 16 2022

a(A003136(n)) > 0; a(A034020(n)) = 0;
a(A118886(n)) > 1; a(A198772(n)) = 1;
a(A198773(n)) = 2; a(A198774(n)) = 3;
a(A198775(n)) = 4;
a(A198799(n)) = n and a(m) <> n for m < A198799(n). - Reinhard Zumkeller, Oct 30 2011, corrected by M. F. Hasler, Mar 05 2018
In the prime factorization of n, let S_1 be the set of distinct prime factors p_i for which p_i == 1 (mod 3), let S_2 be the set of distinct prime factors p_j for which p_j == 2 (mod 3), and let M be the exponent of 3. Then n = 3^M * (Product_{p_i in S_1} p_i ^ e_i) * (Product_{p_j in S_2} p_j ^ e_j), and the number of solutions for x^2 + xy + y^2 = n, 0 <= x <= y is floor((Product_{p_i in S_1} (e_i + 1) + 1) / 2) if all e_j are even and 0 otherwise. E.g. a(1729) = 4 since 1729 = 7^1*13^1*19^1 and floor(((1+1)*(1+1)*(1+1)+1)/2) = 4. - Seth A. Troisi, Jul 02 2020
a(n) = ceiling(A004016(n)/12) = (A002324(n) + A145377(n)) / 2. - Andrey Zabolotskiy, Dec 23 2021

Edited by M. F. Hasler, Mar 05 2018

A195198 Characteristic function of squares or three times squares.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Michael Somos, Sep 11 2011

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

A214284 is a similar sequence except with five instead of three. - Michael Somos, Oct 22 2017

			G.f. = 1 + q + q^3 + q^4 + q^9 + q^12 + q^16 + q^25 + q^27 + q^36 + q^48 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. See p. 133 Theorem 3.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A089801, A127692, A145377, A214284.

a[ n_] := SeriesCoefficient[ Series[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}], {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n] || OddQ [ Length @ Divisors[3 n]]]];
Table[If[AnyTrue[{Sqrt[n],Sqrt[3n]},IntegerQ],1,0],{n,0,110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 22 2020 *)

PARI
```
{a(n) = issquare(n) || issquare(3*n)};
```

{a(n) = if( n<1, n==0, direuler( p=2, n, if( p==3, 1 + X, 1) / (1 - X^2))[n])};

Euler transform of period 12 sequence [1, -1, 1, 0, 0, -1, 0, 0, 1, -1, 1, -1, ...].
Expansion of psi(q^3) * f(-q^2, -q^10) / f(-q, -q^11) in powers of q where psi(), is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
Multiplicative with a(0) = a(3^e) = 1, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) + theta_3(q^3)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(3*k^2)).
Dirichlet g.f.: zeta(2*s) * (1 + 3^-s).
a(n) = A145377(n) unless n=0. a(3*n + 2) = 0. a(2*n + 1) = A127692(n). a(3*n) = a(n). a(3*n + 1) = A089801(n).
Sum_{k=0..n} a(k) ~ (1+1/sqrt(3)) * sqrt(n). - Amiram Eldar, Sep 14 2023

cp's OEIS Frontend

A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

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A088534 Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.

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A195198 Characteristic function of squares or three times squares.

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A157235 Number of primitive inequivalent oblique sublattices of hexagonal (triangular) lattice of index n (equivalence and symmetry of sublattices are determined using only parent lattice symmetries).

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