A060983 - OEIS

A060983

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

1, 7, 13, 35, 31, 91, 57, 154, 130, 217, 133, 455, 183, 399, 403, 644, 307, 910, 381, 1085, 741, 931, 553, 2002, 806, 1281, 1209, 1995, 871, 2821, 993, 2632, 1729, 2149, 1767, 4550, 1407, 2667, 2379, 4774, 1723, 5187, 1893, 4655, 4030, 3871

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OFFSET

1,2

COMMENTS

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, gcd(a,b,c,d,e,f) = 1.

From Álvar Ibeas, Oct 30 2015: (Start)

a(n) is the number of 2-generated subgroups of Z^3 with order n.

The Dirichlet convolution of a(n) with A010057 gives A001001.

Dirichlet convolution of A254981 and A000290.

(End)

LINKS

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sublattices.

FORMULA

From Álvar Ibeas, Oct 30 2015: (Start)

a(n) = Sum_{d^3 | n} mu(d) * A001001(n/d^3).

Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) / zeta(3s). (End)

Sum_{k=1..n} a(k) ~ Pi^2 * zeta(3) * n^3 / (18*zeta(9)). - Vaclav Kotesovec, Feb 01 2019

Multiplicative with a(p) = p^2+p+1, and a(p^e) = p^(e-2)*(p^e + (p^(e-1)-1)/(p-1)) for e >= 2. - Amiram Eldar, Aug 27 2023

MATHEMATICA

f[p_, e_] := (p^2 + p + 1) * If[e == 1, 1, p^(e - 2)*(p^e + (p^(e - 1) - 1)/(p - 1))]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

CROSSREFS

Cf. A001001, with which it agrees unless n is divisible by a cube.

Cf. A000290, A010057, A254981.

Sequence in context: A331960 A061204 A334783 * A001001 A067692 A117706

Adjacent sequences: A060980 A060981 A060982 * A060984 A060985 A060986

KEYWORD

nonn,easy,mult

AUTHOR

N. J. A. Sloane, May 11 2001

STATUS

approved