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A147848
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Number (up to isomorphism) of groups of order 2n that have Z/nZ as a subgroup (that is, that have an element of order n).
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1
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1, 2, 2, 4, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 4, 6, 2, 4, 2, 8, 4, 4, 2, 12, 2, 4, 2, 8, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 4, 12, 2, 8, 2, 8, 4, 4, 2, 12, 2, 4, 4, 8, 2, 4, 4, 12, 4, 4, 2, 16
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| This sequence is related to A060594 : in fact, for every square root of unity modulo n, there are either one or two such groups of order 2n.
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LINKS
| Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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FORMULA
| a(2) = 2; a(2^2) = 4; a(2^k) = 6 for k >= 3.
a(p^k) = 2 for any odd prime number p and k >= 1.
For other values of n, you can find a(n) by using the fact that the sequence is multiplicative.
Dirichlet g.f.: zeta^2(s) *(1+2^s+2^(1-s)-4^(1-s)+6*4^(-s)) / ( zeta(2*s)*(1+2^s) ). - R. J. Mathar, Jun 01 2011
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EXAMPLE
| Two such groups that always exist are the cyclic group Z/(2n)Z and the dihedral group Dih_n. If n is prime, these are the only such groups, so the n-th term equals 2.
For even values of n, we also have the direct product Z/nZ x Z/2Z and the dicyclic group Dic_n. If n = 2p with p prime, there are no other groups, so the n-th term equals 4.
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MATHEMATICA
| a[1] = 1; a[2] = 2; a[4] = 4; a[n_] := a[n] = (f = FactorInteger[n]; np = Length[f]; Which[ np == 1 && f[[1, 1]] == 2 && f[[1, 2]] >= 3, 6, np == 1 && PrimeQ[f[[1, 1]]] && f[[1, 2]] >= 1, 2, np > 1 && f[[1, 1]] != 2, 2^np, np > 1 && f[[1]] == {2, 1}, 2^np, np > 1 && f[[1]] == {2, 2}, 2^(np+1), np > 1 && f[[1, 1]] == 2 && f[[1, 2]] > 1, 3*2^np, True, 0]); Table[a[n], {n, 1, 60}](* From Jean-François Alcover, Nov 22 2011 *)
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PROG
| (PARI) a(n)=my(k=valuation(n, 2)); max(2*min(k, 3), 1)<<omega(n>>k) \\ Charles R Greathouse IV, Nov 22 2011
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CROSSREFS
| Cf. A060594.
Sequence in context: A171991 A118776 A092520 * A193432 A129089 A169594
Adjacent sequences: A147845 A147846 A147847 * A147849 A147850 A147851
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KEYWORD
| easy,nice,nonn,mult
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AUTHOR
| Ilia Smilga (ilia.smilga(AT)ens.fr), Nov 15 2008
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EXTENSIONS
| Extended comments, references and confirmed "mult" keyword. - Ilia Smilga (ilia.smilga(AT)ens.fr), Nov 17 2008
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