cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Álvar Ibeas, Oct 30 2015

Keywords

Comments

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.

Examples

			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

References

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 1024, 29525, 524800, 2441407, 30233600, 47079209, 268698112, 581150417, 2500000768, 2593742461, 15494720000, 11488207655, 48209110016, 72082541675, 137573433856, 125999618779, 595098027008, 340614792101, 1281250393600
Offset: 1

Views

Author

Seiichi Manyama, May 14 2021

Keywords

Comments

Inverse Moebius transform of A160957.

Links

Crossrefs

Cf. A000010, A013668, A160957, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344305.

Programs

  • Mathematica
    f[p_, e_] := 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a160957(n) = sumdiv(n, d, moebius(n/d)*d^10)/eulerphi(n);
a(n) = sumdiv(n, d, a160957(d));

Formula

a(n) = Sum_{x_1|n, x_2|n, ..., x_10|n} phi(x_1)*phi(x_2)* ... *phi(x_10)/phi(lcm(x_1, x_2, ..., x_10)).
If p is prime, a(p) = 1 + (p^10 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)/10) * Product_{p prime} ((1-1/p^9)/(p^2*(1-1/p))) = 0.1944248708... . (End)

A160964 a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 11.

Original entry on oeis.org

2047, 2094081, 60435628, 1072169472, 4997558082, 61825647444, 96371138776, 548950769664, 1189554465924, 5112501917886, 5309390815620, 31654731491328, 23516361067738, 98587674967848, 147547904812968, 281062794067968, 257921219638566, 1216914218640252
Offset: 1

Views

Author

N. J. A. Sloane, Nov 19 2009

Keywords

Links

Crossrefs

Cf. A000010, A013667, A013668, A160957.

Programs

  • Mathematica
    f[p_, e_] := p^(9*e - 9) * (p^10-1) / (p-1); a[1] = 2047; a[n_] := 2047 * Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
  • PARI
    a(n) = {my(f = factor(n)); 2047 * prod(i = 1, #f~, (f[i,1]^10 - 1)*f[i,1]^(9*f[i,2] - 9)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

Formula

From Amiram Eldar, Nov 08 2022: (Start)
a(n) = 2047 * A160957(n).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (2047/10) * Product_{p prime} (1 + (p^9-1)/((p-1)*p^10)) = 397.5922753... .
Sum_{k>=1} 1/a(k) = (zeta(9)*zeta(10)/2047) * Product_{p prime} (1 - 2/p^10 + 1/p^19) = 0.0004890150305... . (End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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