A145394 -id:A145394 - 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

A003051 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 18, 9, 17, 16, 13, 9, 28, 12, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 21, 27, 16, 26, 13, 24, 18, 26, 13, 40, 14

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
From Andrey Zabolotskiy, Mar 10 2018: (Start)
If only primitive sublattices are considered, we get A003050.
Here only rotations and reflections preserving the parent hexagonal lattice are allowed. If reflections are not allowed, we get A145394. If any rotations and reflections are allowed, we get A300651.
In other words, the parent lattice of the sublattices under consideration has Patterson symmetry group p6mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6).
Rutherford says at p. 161 that his sequence for p6mm differs from this sequence, but it seems that with the current definition and terms of this sequence, this actually is his p6mm sequence, and the sequence he thought to be this one is actually A300651. Also, he says that a(n) != A300651(n) only when A002324(n) > 2 (first time happens at n = 49), but actually these two sequences differ at other terms, too, for example, at n = 42 (see illustration). (End)

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217. [Annotated and corrected scanned copy]
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see Table 3).
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024.
W. Kurth, Enumeration of Platonic maps on the torus, Discrete Math. 61 (1986), 71-83.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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 2].
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003050, A054384, A001615, A006984, A054345, A054346, A000203, A069734, A145391, A145392, A145393, A145394, A112689, A159842.

max = 73; A145390 = Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1 - x^n), {n, 1, max}], {x, 0, max}], x], 1]; A002324[n_] := (dn = Divisors[n]; Count[dn, ?(Mod[#, 3] == 1 & )] - Count[dn, ?(Mod[#, 3] == 2 & )]); a[n_] := (DivisorSigma[1, n] + 2 A002324[n] + 3*A145390[[n]])/6; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Oct 11 2011, after given formula *)

a(n) = Sum_{ m^2 | n } A003050(n/m^2).
a(n) = (A000203(n) + 2*A002324(n) + 3*A145390(n))/6. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ d|n } A112689(d+1). - Andrey Zabolotskiy, Aug 29 2019
a(n) = Sum_{ d|n } floor(d/6) + 1 - 1*[d == 2 or 6 (mod 12)] + 1*[d == 4 (mod 12)]. [Kurth] - Brahadeesh Sankarnarayanan, Feb 24 2023

A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

Original entry on oeis.org

1, 2, 2, 4, 3, 5, 3, 7, 5, 7, 4, 11, 5, 8, 8, 12, 6, 13, 6, 15, 10, 11, 7, 21, 10, 13, 12, 18, 9, 22, 9, 21, 14, 16, 14, 29, 11, 17, 16, 29, 12, 28, 12, 25, 23, 20, 13, 39, 16, 27, 20, 29, 15, 34, 20, 36, 22, 25, 16, 50, 17, 26, 29, 38, 24, 40, 18, 36, 26, 40

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

From Andrey Zabolotskiy, Mar 12 2018: (Start)
If reflections are not allowed, we get A145392. If any rotations and reflections are allowed, we get A054346.
The parent lattice of the sublattices under consideration has Patterson symmetry group p4mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145394 (p6), A003051 (p6mm).
Rutherford says at p. 161 that a(n) != A054346(n) only when A002654(n) > 2, but actually these two sequence differ at other terms, too, for example, at n = 30 (see illustration). (End)

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see table 6 and fig. 2).
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 2; beware the typo in a(5).]
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. A054345, A054346, A167156, A008621.
Cf. A000203, A069734, A145391, A145392, A145394, A003051, A002324, A002654, A069735, A145390.
Cf. A019590, A157228, A157226, A157230, A157231, A053866, A025441.

terms = 70;
CoefficientList[Sum[(1/((1-x^m)(1-x^(4m)))-1), {m, 1, terms}] + O[x]^(terms + 1), x] // Rest (* Jean-François Alcover, Aug 05 2018 *)

a(n) = (A000203(n) + A002654(n) + A069735(n) + A145390(n))/4. [Rutherford] - N. J. A. Sloane, Mar 13 2009
G.f.: Sum_{ m>=1 } (1/((1-x^m)(1-x^(4m))) - 1). [Hanany, Orlando & Reffert, eq. (6.8)] - Andrey Zabolotskiy, Jul 05 2017
a(n) = Sum_{ m: m^2|n } A019590(n/m^2) + A157228(n/m^2) + A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2) = A053866(n) + A025441(n) + Sum_{ m: m^2|n } A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2). [Rutherford] - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A008621(d) = Sum_{ d|n } (1 + floor(d/4)). [From the above-given g.f.] - Andrey Zabolotskiy, Jul 17 2019

New name from Andrey Zabolotskiy, Mar 12 2018

A069734 Number of pairs (p,q), 0<=p<=q, such that p+q divides n.

Original entry on oeis.org

1, 3, 3, 6, 4, 9, 5, 11, 8, 12, 7, 19, 8, 15, 14, 20, 10, 24, 11, 26, 18, 21, 13, 37, 17, 24, 22, 33, 16, 42, 17, 37, 26, 30, 26, 53, 20, 33, 30, 52, 22, 54, 23, 47, 42, 39, 25, 71, 30, 51, 38, 54, 28, 66, 38, 67, 42, 48, 31, 94, 32, 51, 55, 70, 44, 78, 35, 68, 50, 78, 37, 108

Offset: 1

Valery A. Liskovets, Apr 07 2002

Also number of orientable coverings of the Klein bottle with 2n lists (orientable m-list coverings exist only for even m).
Equals row sums of triangle A178650. - Gary W. Adamson, May 31 2010
Also number of inequivalent sublattices of index n of the rectangular lattice, that has the p2mm (pmm) symmetry group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - Andrey Zabolotskiy, Mar 12 2018

			There are 9 pairs (p,q), 0<=p<=q, such that p+q divides 6: (0,1), (0,2), (0,3), (0,6), (1,1), (1, 2), (1, 5), (2, 4), (3, 3); thus a(6) = 9.
x + 3*x^2 + 3*x^3 + 6*x^4 + 4*x^5 + 9*x^6 + 5*x^7 + 11*x^8 + 8*x^9 + ...

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
V. A. Liskovets and A. Mednykh, Number of non-orientable coverings of the Klein bottle
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 2]. [From _N. J. A. Sloane_, Feb 23 2009]
Index entries for sequences related to sublattices

Cf. A069733, A069735, A046524, A178650.
Cf. A000203, A145391-A145394, A003051.
Cf. A304182, A304183, A008619, A007503, A183063.

with(numtheory): a := n -> (sigma(n) + tau(n) + `if`(irem(n,2) = 1, 0, tau(n/2)))/2: seq(a(n), n=1..72); # Peter Luschny, Jul 20 2019

a[n_] := (DivisorSigma[1, n] + DivisorSigma[0, n] + If[OddQ[n], 0, DivisorSigma[0, n/2]])/2;
Array[a, 72] (* Jean-François Alcover, Aug 27 2019, from Maple *)

{a(n) = if( n<1, 0, sum( k=1, n, sum( j=0, k, n%(j+k) == 0)))} /* Michael Somos, Mar 24 2012 */

a(n) = A046524(2n) - A069733(2n).
Inverse Moebius transform of: 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... G.f.: Sum_{n>0} x^n*(1+x^n-x^(2*n))/(1-x^(2*n))/(1-x^n). - Vladeta Jovovic, Feb 03 2003
a(n) = (A000203(n) + A069735(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A304182(n/m^2) + A304183(n/m^2) = A069735(n) + Sum_{ m: m^2|n } A304183(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A008619(d) = Sum_{ d|n } (1 + floor(d/2)). - Andrey Zabolotskiy, Jul 20 2019
a(n) = (A007503(n) + A183063(n))/2. - Peter Luschny, Jul 20 2019

New description from Vladeta Jovovic, Feb 03 2003

A145392 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/2 to give the other.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 4, 8, 7, 10, 6, 14, 8, 12, 12, 16, 10, 20, 10, 22, 16, 18, 12, 30, 17, 22, 20, 28, 16, 36, 16, 32, 24, 28, 24, 46, 20, 30, 28, 46, 22, 48, 22, 42, 40, 36, 24, 62, 29, 48, 36, 50, 28, 60, 36, 60, 40, 46, 30, 84, 32, 48, 52, 64, 44, 72, 34, 64, 48, 72

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

From Andrey Zabolotskiy, Mar 12 2018: (Start)
The parent lattice of the sublattices under consideration has Patterson symmetry group p4, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145393 (p4mm), A145394 (p6), A003051 (p6mm).
If we count sublattices related by parent-lattice-preserving reflection as equivalent, we get A145393 instead of this sequence. If we count sublattices related by rotation of the sublattice only (but not parent lattice; equivalently, sublattices related by rotation by angle which is not a multiple of Pi/2; see illustration in links) as equivalent, we get A054345. If we count sublattices related by any rotation or reflection as equivalent, we get A054346.
Rutherford says at p. 161 that a(n) != A054345(n) only when A002654(n) > 1, but actually these two sequences differ at other terms, too, for example, at n = 15 (see illustration). (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
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 2; beware the typo in a(13).]
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. A000203, A002654, A069734, A145391, A145393, A145394, A003051, A054345, A054346, A000089, A157224, A004525.

A002654(n) = sumdiv(n, d, (d%4==1) - (d%4==3));
A145392(n) = ((sigma(n) + A002654(n))/2); \\ Antti Karttunen, Nov 23 2017

a(n) = (A000203(n) + A002654(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A000089(n/m^2) + A157224(n/m^2) = A002654(n) + Sum_{ m: m^2|n } A157224(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A004525(d). - Andrey Zabolotskiy, Aug 29 2019

New name from Andrey Zabolotskiy, Mar 12 2018

A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 5, 10, 8, 10, 7, 17, 8, 13, 14, 19, 10, 21, 11, 24, 18, 19, 13, 35, 17, 22, 22, 31, 16, 38, 17, 36, 26, 28, 26, 50, 20, 31, 30, 50, 22, 50, 23, 45, 42, 37, 25, 69, 30, 48, 38, 52, 28, 62, 38, 65, 42, 46, 31, 90, 32, 49, 55, 69, 44, 74, 35, 66, 50, 74

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

The centered rectangular lattice has symmetry group c2mm, or cmm. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - Andrey Zabolotskiy, Mar 12 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] [see Table 8].
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 2.]
Index entries for sequences related to sublattices

Cf. A000203, A069734, A145390, A145392, A145393, A145394, A003051, A060594, A157223, A004525.

a060594[n_] := Switch[Mod[n, 8], 2|6, 2^(PrimeNu[n] - 1), 1|3|4|5|7, 2^PrimeNu[n], 0, 2^(PrimeNu[n] + 1)];
a145390[n_] := Sum[If[IntegerQ[Sqrt[d]], a060594[n/d], 0], {d, Divisors[n]} ];
a[n_] := (DivisorSigma[1, n] + a145390[n])/2;
Array[a, 100] (* Jean-François Alcover, Aug 31 2018 *)

a(n) = (A000203(n) + A145390(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A060594(n/m^2) + A157223(n/m^2) = A145390(n) + Sum_{ m: m^2|n } A157223(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A004525(d+1). - Andrey Zabolotskiy, Aug 29 2019

New name from Andrey Zabolotskiy, Mar 12 2018

New name from Andrey Zabolotskiy, Jan 19 2022

A054384 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated to give the other.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 3, 5, 5, 6, 4, 10, 5, 7, 8, 11, 6, 13, 7, 14, 10, 12, 8, 20, 11, 13, 14, 17, 10, 24, 11, 21, 16, 18, 14, 31, 13, 19, 18, 30, 14, 28, 15, 28, 26, 24, 16, 42, 17, 31, 24, 31, 18, 40, 24, 35, 26, 30, 20, 56, 21, 31, 31, 43, 26, 48, 23, 42, 32, 42, 24, 65

Offset: 0

N. J. A. Sloane, May 08 2000

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

If reflections are allowed, we get A300651. If only rotations that preserve the parent hexagonal lattice are allowed, we get A145394. The analog for square lattice is A054345. - Andrey Zabolotskiy, Mar 10 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1000
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.1 for the Dirichlet g.f. zeta^SL_{x^2+xy+y^2}(s).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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 2]. - From _N. J. A. Sloane_, Feb 23 2009
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003051, A054345, A054346, A145394, A300651.

# see A159842 for the definitions of dc, fin, u, N
def gg(m, k1, minus = True):
    def f(n):
        if n == 1: return 1
        r = 1
        for (p, k) in factor(n):
            if p % 3 != m or k1 and k > 1: return 0
            if minus: r *= (-1)**k
        return r
    return f
g1, g2, g3 = gg(1, True), gg(1, True, False), gg(2, False)
def a_SL(n):
    return (dc(u, N, g1)(n) + 2 * dc(u, g3)(n)) / 3
print([a_SL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

A300651 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if they are related by any rotation or reflection.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 17, 9, 17, 16, 13, 9, 28, 11, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 20, 27, 16, 26, 13, 24, 18, 24, 13, 40

Offset: 1

Andrey Zabolotskiy, Mar 10 2018

nonn

If we count sublattices as equivalent only if they are related by a rotation, we get A054384 instead of this sequence. If we only allow rotations and reflections that preserve the parent (hexagonal) lattice, we get A003051; the first discrepancy is at n = 42 (see illustration), the second is at n = 49. If both restrictions are applied, i.e., only rotations preserving the parent lattice are allowed, we get A145394. The analog for square lattice is A054346.

Although A003051 has its counterpart A003050 which counts primitive sublattices only, this sequence has no such counterpart sequence because a primitive sublattice can turn to a non-primitive one via a non-parent-lattice-preserving rotation, so the straightforward definition of primitiveness does not work in this case.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024. See section 6.1 for the Dirichlet g.f. zeta^GL_{x^2+xy+y^2}(s).
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003050, A003051, A054384, A145394, A054346.

# See A159842 and A054384 for the definitions of functions used here
def a_GL(n):
    return (a_SL(n) + dc(fin(1, -1, 0, 2), u, u, g2)(n)) / 2
print([a_GL(n) for n in range(1, 100)]) # Andrey Zabolotskiy, Sep 22 2024

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

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A003051 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.

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A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

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A069734 Number of pairs (p,q), 0<=p<=q, such that p+q divides n.

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A145392 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/2 to give the other.

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A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.

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