A160870 -id:A160870 - cp's OEIS frontend

A160884 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

Original entry on oeis.org

3, 15, 121, 1271, 15233

Offset: 1

N. J. A. Sloane, Nov 15 2009

nonn

A160885 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

Original entry on oeis.org

4, 29, 281, 3231, 42099

Offset: 1

N. J. A. Sloane, Nov 15 2009

nonn

A160886 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

Original entry on oeis.org

4, 26, 250, 3086, 44674

Offset: 1

N. J. A. Sloane, Nov 15 2009

nonn

A160887 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

Original entry on oeis.org

4, 35, 431, 6267, 102555

Offset: 1

N. J. A. Sloane, Nov 15 2009

nonn

A160888 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

Original entry on oeis.org

3, 19, 225, 3475, 61521

Offset: 1

N. J. A. Sloane, Nov 15 2009

nonn

A001001 Number of sublattices of index n in generic 3-dimensional lattice.

Original entry on oeis.org

1, 7, 13, 35, 31, 91, 57, 155, 130, 217, 133, 455, 183, 399, 403, 651, 307, 910, 381, 1085, 741, 931, 553, 2015, 806, 1281, 1210, 1995, 871, 2821, 993, 2667, 1729, 2149, 1767, 4550, 1407, 2667, 2379, 4805, 1723, 5187, 1893, 4655, 4030, 3871, 2257, 8463, 2850, 5642, 3991, 6405, 2863

Offset: 1

N. J. A. Sloane

These sublattices are in 1-1 correspondence with matrices
[a b d]
[0 c e]
[0 0 f]
with acf = n, b = 0..c-1, d = 0..f-1, e = 0..f-1. The sublattice is primitive if gcd(a,b,c,d,e,f) = 1.
Total area of all distinct rectangles whose side lengths are divisors of n, and whose length is an integer multiple of the width. - Wesley Ivan Hurt, Aug 23 2020

Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(d), pp. 76 and 113.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Michael Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.; arXiv:math/0605222 [math.MG], 2006.
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
J. Liouville, Théorème concernant les sommes de diviseurs des nombres, Journal de mathématiques pures et appliquées 2e série, tome 2 (1857), p. 56.
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. (1992) A48, 500-508
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [_N. J. A. Sloane_, Mar 14 2009]
Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.
Index entries for sequences related to sublattices.

Column 3 of A160870.
Cf. A060983, A064987 (Mobius transform).
Cf. A061256, A226313, A301777.
Primes in this sequence are in A053183.
Cf. A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

a001001 n = sum [sum [k * (if k `mod` l == 0 then l else 0) | k <- [1..n], n `mod` k == 0 ] | l <- [1..n]]
a = [ a001001 n | n <- [1..53]]
putStrLn $ concat $ map (++ ", ") (map show a) -- Miles Wilson, Apr 04 2025

nmax := 100:
L12 := [seq(1,i=1..nmax) ];
L27 := [seq(i,i=1..nmax) ];
L290 := [seq(i^2,i=1..nmax) ];
DIRICHLET(L12,L27) ;
DIRICHLET(%,L290) ; # R. J. Mathar, Sep 25 2017

a[n_] := Sum[ d*DivisorSigma[1, d], {d, Divisors[n]}]; Table[ a[n], {n, 1, 42}] (* Jean-François Alcover, Jan 20 2012, after Vladeta Jovovic *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 2}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j);
t=1/prod(j=1,N, eta(x^(j))^j)
t=log(t)
t=serconvol(t,c)
Vec(t)
/* Joerg Arndt, May 03 2008 */

a(n)=sumdiv(n,d, d * sumdiv(d,t, t ) );  /* Joerg Arndt, Oct 07 2012 */

a(n)=sumdivmult(n,d, sigma(d)*d) \\ Charles R Greathouse IV, Sep 09 2014

If n = Product p^m, a(n) = Product (p^(m + 1) - 1)(p^(m + 2) - 1)/(p - 1)(p^2 - 1). Or, a(n) = Sum_{d|n} sigma(n/d)*d^2, Dirichlet convolution of A000290 and A000203.
a(n) = Sum_{d|n} d*sigma(d). - Vladeta Jovovic, Apr 06 2001
Multiplicative with a(p^e) = ((p^(e+1)-1)(p^(e+2)-1))/((p-1)(p^2-1)). - David W. Wilson, Sep 01 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
a(n) = Sum_{d1|n, d2|n, d1|d2} d1*d2. - Wesley Ivan Hurt, Aug 23 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2*zeta(3)/18 = 0.659101... . - Amiram Eldar, Oct 19 2022
G.f.: Sum_{k>=1} Sum {l>=1} k*l^2*x^(k*l - 1)/(1 - x^(k*l)). - Miles Wilson, Apr 04 2025

A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 13, 15, 1, 1, 6, 28, 40, 31, 1, 1, 12, 31, 120, 121, 63, 1, 1, 8, 91, 156, 496, 364, 127, 1, 1, 12, 57, 600, 781, 2016, 1093, 255, 1, 1, 12, 112, 400, 3751, 3906, 8128, 3280, 511, 1, 1, 18, 117, 960, 2801, 22932, 19531, 32640

Offset: 1

Álvar Ibeas, Oct 30 2015

All the enumerated lattices have full rank k, since the quotient group is finite.
For m>=1, T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_nm x (C_m)^(k-1); and also, (C_nm)^(k-1) x C_m.
Also, number of subgroups of (C_n)^k isomorphic to C_n (and also, to (C_n)^{k-1}), cf. [Butler, Lemma 1.4.1].
T(n,k) is the sum of the divisors d of n^(k-1) such that n^(k-1)/d is k-free. Namely, the coefficient in n^(-(k-1)*s) of the Dirichlet series zeta(s) * zeta(s-1) / zeta(ks).
Also, number of isomorphism classes of connected (C_n)-fold coverings of a connected graph with circuit rank k.
Columns are multiplicative functions.

			There are 7 = A160870(4,2) lattices of volume 4 in Z^2. Among them, only one (<(2,0), (0,2)>) gives the quotient group C_2 x C_2, whereas the rest give C_4. Hence, T(4,2) = 6 and T(1,2) = 1.
Array begins:
      k=1    k=2    k=3    k=4    k=5    k=6
n=1     1      1      1      1      1      1
n=2     1      3      7     15     31     63
n=3     1      4     13     40    121    364
n=4     1      6     28    120    496   2016
n=5     1      6     31    156    781   3906
n=6     1     12     91    600   3751  22932

Lynne M. Butler, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., Vol. 112, No. 539, 1994.

Álvar Ibeas, First 100 antidiagonals, flattened
Jin Ho Kwak, Jang-Ho Chun, and Jaeun Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285. [Page 277]
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Examples 2 and 3]

Rows (1-11): A000012, A000225, A003462, A006516, A003463, A160869, A023000, A016152, A016142(n-1), A046915(n-1), A016123(n-1).
Columns (1-17): A000012, A001615, A160889, A160891, A160893, A160895, A160897, A160908, A160953, A160957, A160960, A160972, A161010, A161025, A161139, A161167, A161213.
Main diagonal gives A344210.
Cf. A000203, A008683, A160870.

f[p_, e_, k_] := p^((k - 1)*(e - 1))*(p^k - 1)/(p - 1); T[n_, 1] = T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 08 2022 *)

T(n,k) = J_k(n) / J_1(n) = (Sum_{d|n} mu(n/d) * d^k) / phi(n).
T(n,k) = n^(k-1) * Product_{p|n, p prime} (p^k - 1) / ((p - 1) * p^(k-1)).
Dirichlet g.f. of k-th column: zeta(s-k+1) * Product_{p prime} (1 + p^(-s) + p^(1-s) + ... + p^(k-2-s)).
If n is squarefree, T(n,k) = A160870(n,k) = A000203(n^(k-1)).
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{i=1..n} T(i, k) ~ c * n^k, where c = (1/k) * Product_{p prime} (1 + (p^(k-1)-1)/((p-1)*p^k)).
Sum_{i>=1} 1/T(i, k) = zeta(k-1)*zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(2*k-1)), for k > 2. (End)
T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^k). - Ridouane Oudra, Apr 03 2025

A038994 Number of sublattices of index n in generic 7-dimensional lattice.

Original entry on oeis.org

1, 127, 1093, 10795, 19531, 138811, 137257, 788035, 896260, 2480437, 1948717, 11798935, 5229043, 17431639, 21347383, 53743987, 25646167, 113825020, 49659541, 210837145, 150021901, 247487059, 154764793, 861322255, 317886556, 664088461, 678468820, 1481689315

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sublattices.

Column 7 of A160870.

Cf. A001001, A038991, A038992, A038993, A038995, A038996, A038997, A038998, A038999.

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 6}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=7.
Multiplicative with a(p^e) = Product_{k=1..6} (p^(e+k)-1)/(p^k-1).
Sum_{k=1..n} a(k) ~ c * n^7, where c = Pi^12*zeta(3)*zeta(5)*zeta(7)/3572100 = 0.325206... . - Amiram Eldar, Oct 19 2022

More terms from Amiram Eldar, Aug 29 2019

A038991 Number of sublattices of index n in generic 4-dimensional lattice.

Original entry on oeis.org

1, 15, 40, 155, 156, 600, 400, 1395, 1210, 2340, 1464, 6200, 2380, 6000, 6240, 11811, 5220, 18150, 7240, 24180, 16000, 21960, 12720, 55800, 20306, 35700, 33880, 62000, 25260, 93600, 30784, 97155, 58560, 78300, 62400, 187550, 52060, 108600, 95200, 217620, 70644, 240000, 81400

Offset: 1

N. J. A. Sloane

M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)
M. Baake and U. Grimm, Combinatorial problems of (quasi)crystallography, arXiv:math-ph/0212015, 2002.
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.
Index entries for sequences related to sublattices

Column 4 of A160870.

Cf. A001001, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

a[n_] := DivisorSum[n, #*DivisorSum[#, #*DivisorSum[#, #&]&]&]; Array[a, 50] (* Jean-François Alcover, Dec 02 2015, after Joerg Arndt *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 3}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z ) ) ); /* Joerg Arndt, Oct 07 2012 */

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=4.
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3).
Dirichlet convolution of A000578 and A001001.
Multiplicative with a(p^e) = Product_{k=1..3} (p^(e+k)-1)/(p^k-1).
Sum_{k=1..n} a(k) ~ Pi^6 * Zeta(3) * n^4 / 2160. - Vaclav Kotesovec, Feb 01 2019
Conjectured g.f.: Sum_{k>=1} Sum {l>=1} Sum {m>=1} k*l^2*m^3*x^(k*l*m)/(1 - x^(k*l*m)) (by extension of g.f for A001001). - Miles Wilson, Apr 05 2025

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Joerg Arndt, Oct 07 2012

A038992 Number of sublattices of index n in generic 5-dimensional lattice.

Original entry on oeis.org

1, 31, 121, 651, 781, 3751, 2801, 11811, 11011, 24211, 16105, 78771, 30941, 86831, 94501, 200787, 88741, 341341, 137561, 508431, 338921, 499255, 292561, 1429131, 508431, 959171, 925771, 1823451, 732541, 2929531, 954305, 3309747, 1948705, 2750971, 2187581, 7168161, 1926221

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
Index entries for sequences related to sublattices.

Column 5 of A160870.

Cf. A001001, A038991, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

a[n_] := DivisorSum[n, #*DivisorSum[#, #*DivisorSum[#, #*DivisorSum[#, # &] &] &] &]; Array[a, 50] (* Jean-François Alcover, Dec 02 2015, after Joerg Arndt *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 4}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z * sumdiv(z,w, w ) ) ) ); /* Joerg Arndt, Oct 07 2012 */

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=5.
Multiplicative with a(p^e) = Product_{k=1..4} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3)*zeta(s-4). Dirichlet convolution of A038991 with A000583. - R. J. Mathar, Mar 31 2011
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^6*zeta(3)*zeta(5)/2700 = 0.443822... . - Amiram Eldar, Oct 19 2022

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Joerg Arndt, Oct 07 2012

cp's OEIS Frontend

A160884 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

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A160885 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

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A160886 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

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A160887 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

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A160888 Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition.

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A001001 Number of sublattices of index n in generic 3-dimensional lattice.

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A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

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A038994 Number of sublattices of index n in generic 7-dimensional lattice.

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A038991 Number of sublattices of index n in generic 4-dimensional lattice.

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A038992 Number of sublattices of index n in generic 5-dimensional lattice.

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