A304182 -id:A304182 - cp's OEIS frontend

A157220 Erroneous version of A304182.

Original entry on oeis.org

3, 2, 4, 2, 6, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 2, 6, 2, 8, 4, 6, 2, 6, 2, 6, 2, 4, 2, 12

Offset: 2

N. J. A. Sloane, Feb 25 2009

dead

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 4. Contains errors for n = 24 and 28.]

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

A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.

Original entry on oeis.org

1, 3, 2, 5, 2, 6, 2, 7, 3, 6, 2, 10, 2, 6, 4, 9, 2, 9, 2, 10, 4, 6, 2, 14, 3, 6, 4, 10, 2, 12, 2, 11, 4, 6, 4, 15, 2, 6, 4, 14, 2, 12, 2, 10, 6, 6, 2, 18, 3, 9, 4, 10, 2, 12, 4, 14, 4, 6, 2, 20, 2, 6, 6, 13, 4, 12, 2, 10, 4, 12, 2, 21, 2, 6, 6, 10, 4, 12, 2, 18, 5, 6, 2, 20, 4, 6, 4, 14, 2, 18

Offset: 1

Valery A. Liskovets, Apr 07 2002

Dirichlet convolution of A000012 by A040001. - R. J. Mathar, Mar 30 2011

a(n) is the number of full-dimensional lattices with volume n in Z^2 which are symmetric about a coordinate axis (equivalently, about both). - Álvar Ibeas, Mar 19 2021

			x + 3*x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 3*x^9 + 6*x^10 + ...

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Valery A. Liskovets and Alexander Mednykh, Number of non-orientable coverings of the Klein bottle, 2002.
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

Equals row sums of triangle A143110. - Gary W. Adamson, Jul 25 2008

Cf. A000005, A000012, A001227, A001620, A040001, A059841, A069734, A099774, A183063, A304182, A320111.

read("transforms") : nmax := 100 :
L := [1,1,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017
with(NumberTheory): seq(tau(n) + `if`(n::odd, 0, tau(n/2)), n=1..100); # Peter Luschny, Mar 19 2021

d[n_] := DivisorSigma[0, n];
a[n_] := If[EvenQ[n], d[n] + d[n/2], d[n]];
Array[a, 100] (* Jean-François Alcover, Aug 27 2019 *)

{a(n) = if( n<1, 0, numdiv(n) + if( n%2, 0, numdiv( n / 2)))} /* Michael Somos, Mar 24 2012 */

Multiplicative with a(2^e)=2e+1 and a(p^e)=e+1 for e>0 and an odd prime p.
a(n) = d(n)+d(n/2) for even n and a(n) = d(n) otherwise where d(n) is the number of divisors of n (A000005).
G.f.: Sum_{k>0} x^k*(1+2*x^k)/(1-x^(2*k)). - Vladeta Jovovic, Dec 16 2002
Dirichlet g.f.: (1+2^(-s))*zeta^2(s) [ Rutherford]. - N. J. A. Sloane, Feb 23 2009
Moebius transform is period 2 sequence [ 1, 2, ...]. - Michael Somos, Mar 24 2012
a(2*n - 1) = A099774(n).
a(n) = Sum_{ m: m^2|n } A304182(n/m^2). - Andrey Zabolotskiy, May 07 2018
Sum_{k=1..n} a(k) ~ 3*n*log(n)/2 + (3*gamma - 3/2 - log(2)/2)*n, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 04 2019
a(n) = 3*tau(n) - tau(2*n). - Ridouane Oudra, Mar 15 2021
a(n) = A320111(n) + (A059841(n)*A000005(n)), i.e. a(n) = A320111(n) if n is odd, and a(n) = A320111(n) + A000005(n) if n is even. - Antti Karttunen, Mar 17 2021
a(n) = A000005(n) + A183063(n) = 2*A000005(n) - A001227(n). - Amiram Eldar, Dec 22 2023

Corrected by T. D. Noe, Nov 13 2006

A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

From Andrey Zabolotskiy, May 09 2018: (Start)
Also, the number of partitions of n into 2 distinct coprime squares.
All such sublattices (including non-primitive ones) are counted in A025441.
The primitive sublattices that have the same symmetries (including the orientation of the mirrors) as the parent lattice are not counted here; they are counted in A019590, and all such sublattices (including non-primitive ones) are counted in A053866.
For n > 2, equals A193138. (End)

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
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 5.]

Cf. A193138, A145393 (all sublattices of the square lattice), A025441, A019590, A053866, A157226, A157230, A157231, A000089, A304182, A224450, A224770, A281877, A024362.

a(n) = (A000089(n) - A019590(n)) / 2. - Andrey Zabolotskiy, May 09 2018
a(n) = 1 if n>2 is in A224450, a(n) = 2 if n is in A224770, a(n) is a higher power of 2 if n is in A281877 (first time reaches 8 at n = 32045). - Andrey Zabolotskiy, Sep 30 2018
a(n) = b(n) for odd n, a(n) = b(n/2) for even n, where b(n) = A024362(n). - Andrey Zabolotskiy, Jan 23 2022

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 4, 1, 1, 1, 4, 2, 1, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

After a(2), this matches A034380 except for n = 63, 65, 80, 85, ... - R. J. Mathar, Feb 27 2009 [Updated by Andrey Zabolotskiy, May 09 2018]

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
J. S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Act. Cryst. A65 (2009) 156-163, Table 5 symmetry *mm2.

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157231, A304182, A060594, A046072, A033948, A272592.

a[n_] := If[n <= 2, 0, Sum[Boole[Mod[k^2, n] == 1], {k, 1, n}]/2];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 12 2023 *)

From Andrey Zabolotskiy, Sep 30 2018: (Start)
a(n) = (A060594(n) - A019590(n))/2.
a(n) = 2^(A046072(n)-1) for n>2. Thus a(n) = 1 if n>2 is in A033948, a(n) = 2 if n is in A272592, etc. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 2, 2, 1, 3, 1, 4, 2, 3, 1, 4, 1, 3, 1, 4, 1, 6, 1, 2, 2, 3, 2, 4, 1, 3, 2, 4, 1, 6, 1, 4, 2, 3, 1, 4, 1, 3, 2, 4, 1, 3, 2, 4, 2, 3, 1, 8, 1, 3, 2, 2, 2, 6, 1, 4, 2, 6, 1, 4, 1, 3, 2, 4, 2, 6, 1, 4, 1, 3, 1, 8, 2, 3, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

Andrey Zabolotskiy's new formula confirms that a(n) indeed is a function of A305891(n). - Antti Karttunen, Oct 01 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
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 5.]

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157230, A157231, A304182, A007875, A029744.

A007875(n) = eulerphi(2^omega(n));
A157226(n) = if(n<=2,n-1,(A007875(n) + if(!(n%2),A007875(n/2)))); \\ Antti Karttunen, Oct 01 2018

From Andrey Zabolotskiy, Sep 30 2018: (Start)
Let b(n) = A007875(n) for n>1, b(1) = 0. Then
a(n) = b(n) for odd n,
a(n) = b(n) + b(n/2) for even n.
Thus the sorted list of all terms (except for a(1)=0) is A029744. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157231 Number of primitive inequivalent oblique sublattices of square lattice of index n (equivalence and symmetry of sublattices are determined using parent lattice symmetries).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 3, 7, 4, 6, 6, 7, 5, 8, 6, 8, 8, 9, 6, 14, 7, 10, 10, 11, 10, 15, 8, 13, 12, 14, 9, 20, 10, 15, 16, 16, 11, 20, 13, 20, 16, 18, 12, 25, 16, 20, 18, 20, 14, 30, 14, 22, 22, 22, 18, 32, 16, 24, 22, 32, 17, 32, 17

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
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 5.]

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157230, A001615, A304182.

a(n) = (A001615(n) - A019590(n) - 2 * (A157228(n) + A157226(n) + A157230(n))) / 4. - Andrey Zabolotskiy, May 09 2018

New name and more terms from Andrey Zabolotskiy, May 09 2018

A304183 Number of primitive inequivalent oblique sublattices of rectangular lattice of index n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 5, 8, 6, 9, 10, 10, 8, 15, 9, 14, 14, 15, 11, 20, 14, 18, 17, 20, 14, 30, 15, 22, 22, 24, 22, 32, 18, 27, 26, 32, 20, 42, 21, 32, 34, 33, 23, 44, 27, 42, 34, 38, 26, 51, 34, 44, 38, 42, 29, 64, 30, 45, 46, 46, 40, 66, 33, 50, 46

Offset: 1

Andrey Zabolotskiy, May 07 2018

nonn

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 4. Contains errors for n = 24 and 28.]

Cf. A304182 (primitive mirror-symmetric sublattices), A069734 (all sublattices), A001615 (primitive sublattices of oblique lattice).

a(n) = (A001615(n) - A304182(n))/2.

A157223 Number of primitive inequivalent oblique sublattices of centered rectangular lattice of index n.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 3, 4, 5, 8, 5, 10, 6, 11, 10, 10, 8, 17, 9, 16, 14, 17, 11, 20, 14, 20, 17, 22, 14, 34, 15, 22, 22, 26, 22, 34, 18, 29, 26, 32, 20, 46, 21, 34, 34, 35, 23, 44, 27, 44, 34, 40, 26, 53, 34, 44, 38, 44, 29, 68, 30, 47, 46, 46, 40, 70, 33, 52, 46

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

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 4.]

Cf. A060594 (primitive mirror-symmetric sublattices), A145390 (all mirror-symmetric sublattices), A145391 (all sublattices), A001615, A304182.

a(n) = (A001615(n) - A060594(n))/2. - Andrey Zabolotskiy, May 09 2018

New name and more terms from Andrey Zabolotskiy, May 09 2018

A157224 Number of primitive inequivalent (up to Pi/2 rotation) nonsquare sublattices of square lattice of index n.

Original entry on oeis.org

0, 1, 2, 3, 2, 6, 4, 6, 6, 8, 6, 12, 6, 12, 12, 12, 8, 18, 10, 18, 16, 18, 12, 24, 14, 20, 18, 24, 14, 36, 16, 24, 24, 26, 24, 36, 18, 30, 28, 36, 20, 48, 22, 36, 36, 36, 24, 48, 28, 44, 36, 42, 26, 54, 36, 48, 40, 44, 30, 72, 30, 48, 48, 48, 40, 72, 34, 54

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

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 4.]

Cf. A000089 (primitive square sublattices), A002654 (all square sublattices), A145392 (all sublattices), A001615, A304182.

a(n) = (A001615(n) - A000089(n))/2. - Andrey Zabolotskiy, May 09 2018

New name and more terms from Andrey Zabolotskiy, May 09 2018

cp's OEIS Frontend

A157220 Erroneous version of A304182.

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

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A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.

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A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.

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A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

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A157226 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the sides of the unit cell of the parent lattice of index n.

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A157231 Number of primitive inequivalent oblique sublattices of square lattice of index n (equivalence and symmetry of sublattices are determined using parent lattice symmetries).

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A304183 Number of primitive inequivalent oblique sublattices of rectangular lattice of index n.

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