A160893 -id:A160893 - cp's OEIS frontend

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.

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

A344219 Number of cyclic subgroups of the group (C_n)^5, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 32, 122, 528, 782, 3904, 2802, 8464, 9923, 25024, 16106, 64416, 30942, 89664, 95404, 135440, 88742, 317536, 137562, 412896, 341844, 515392, 292562, 1032608, 488907, 990144, 803804, 1479456, 732542, 3052928, 954306, 2167056, 1964932, 2839744, 2191164, 5239344, 1926222

Offset: 1

Seiichi Manyama, May 12 2021

Inverse Moebius transform of A160893.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013663, A060648, A064969, A160893, A280184.

f[p_, e_] := 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, sumdiv(n, m, eulerphi(i)*eulerphi(j)*eulerphi(k)*eulerphi(l)*eulerphi(m)/eulerphi(lcm([i, j, k, l, m])))))));

a160893(n) = sumdiv(n, d, moebius(n/d)*d^5)/eulerphi(n);
a(n) = sumdiv(n, d, a160893(d));

a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} phi(x_1)*phi(x_2)*phi(x_3)*phi(x_4)*phi(x_5)/phi(lcm(x_1, x_2, x_3, x_4, x_5)).
If p is prime, a(p) = 1 + (p^5 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)).
Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3939461744... . (End)

A220555 T(n,k) = maximal order N of cyclic group {D,D^2,...,D^N} generated by an n X n Danzer matrix D over Z/kZ, where D is from the m-th Danzer basis and m=2*n+1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 8, 1, 1, 7, 26, 6, 1, 1, 31, 18, 14, 20, 1, 1, 63, 121, 14, 62, 24, 1, 1, 15, 26, 62, 62, 182, 16, 1, 1, 15, 24, 126, 781, 126, 42, 12, 1, 1, 511, 1640, 30, 24, 3751, 114, 28, 24, 1, 1, 63, 9841, 30, 20, 1638, 2801, 28, 78, 60, 1

Offset: 1

L. Edson Jeffery, Dec 15 2012

For definition of Danzer matrix see [Jeffery] (notation differs there!).
Conjecture 1. Let F_n(x)=sum_{j=0..n} A187660(n,j)*x^{(n-1)*j}. Let f_n in Z[x] be any polynomial in x of degree d such that 0<=d<=(n-1)*(n-2). Then the sequence of coefficients of the series expansion of f_n(x)/F_n(x), when taken over Z/kZ, is periodic with period p <= (n-1)*A220555(n,k), for all n,k > 1. (Cf. [Coleman, et al.] for the case for n=2 (generalized Fibonacci).)
Conjecture 2. If G a cyclic multiplicative group generated by an n X n integer matrix over Z/kZ, then |G|<=T(r,k), for some r<=n.
Definition. If T(n,k)>=(k^n-1)/(k-1), for some k>1, then T(n,k) is said to be "optimal."
Conjecture 3. If T(n,k) is optimal, then n is a Queneau number (A054639).
Sequence is read from antidiagonals of array T which begins as
.1...1....1....1......1.......1......1....1.....1.........1
.1...3....8....6.....20......24.....16...12....24........60
.1...7...26...14.....62.....182.....42...28....78.......434
.1...7...18...14.....62.....126....114...28....54.......434
.1..31..121...62....781....3751...2801..124...363.....24211
.1..63...26..126.....24....1638..13072..252....78.......504
.1..15...24...30.....20.....120....400...60....72........60
.1..15.1640...30..32552....4920.240200...60..4920....488280
.1.511.9841.1022.488281.5028751....342.2044.29523.249511591
.1..63...78..126....124....1638.....42..252...234......7812
Rows might be related to Jordan totient functions J_n(k), however, some entries T(n,k) are products of factors of the form (j^n-1)/(j-1).

D. A. Coleman et al., Periods of (q,r)-Fibonacci sequences and Elliptic Curves, Fibonacci Quart. 44, no 1 (2006) 59-70.
L. E. Jeffery, Danzer matrices.

Cf. A001175 (possibly = row 2), A086839 (possibly = column 2), A160893, A160895, A160897, A160960, A160972, A161010, A161025, A161139, A161167, A161213.

Cf. A187772 (gives maximal periods p of Conjecture 1).

A160896 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 6.

Original entry on oeis.org

63, 1953, 7623, 31248, 49203, 236313, 176463, 499968, 617463, 1525293, 1014615, 3781008, 1949283, 5470353, 5953563, 7999488, 5590683, 19141353, 8666343, 24404688, 21352023, 31453065, 18431343, 60496128, 30751875, 60427773, 50014503, 87525648, 46150083, 184560453

Offset: 1

N. J. A. Sloane, Nov 19 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
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. See p. 134.

Cf. A000010, A013662, A013663, A160893.

f[p_, e_] := p^(4*e - 4)*(1 + p + p^2 + p^3 + p^4); a[1] = 63; a[n_] := 63 * Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

a(n) = {my(f = factor(n)); 63 * prod(i = 1, #f~, (f[i,1]^4 + f[i,1]^3 + f[i,1]^2 + f[i,1] + 1)*f[i,1]^(4*f[i,2] - 4));} \\ Amiram Eldar, Nov 08 2022

a(n) = 63*A160893(n). - R. J. Mathar, Mar 16 2016
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^5, where c = (63/5) * Product_{p prime} (1 + (p^4-1)/((p-1)*p^5)) = 23.9347523175... .
Sum_{k>=1} 1/a(k) = (zeta(4)*zeta(5)/63) * Product_{p prime} (1 - 2/p^5 + 1/p^9) = 0.01658573169... . (End)

cp's OEIS Frontend

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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A344219 Number of cyclic subgroups of the group (C_n)^5, where C_n is the cyclic group of order n.

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A220555 T(n,k) = maximal order N of cyclic group {D,D^2,...,D^N} generated by an n X n Danzer matrix D over Z/kZ, where D is from the m-th Danzer basis and m=2*n+1.

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A160896 a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 6.

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Mathematica

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Formula

A160896 a(n) = ((2^b-1)/phi(n))Sum_{d|n} Moebius(n/d)d^(b-1) for b = 6.