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A060983
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Number of primitive sublattices of index n in generic 3-dimensional lattice.
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2
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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
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1,2
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COMMENTS
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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.
a(n) is the number of 2-generated subgroups of Z^3 with order n.
(End)
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A001001, with which it agrees unless n is divisible by a cube.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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