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A060724
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Number of subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).
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8
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1, 5, 6, 15, 8, 30, 10, 37, 23, 40, 14, 90, 16, 50, 48, 83, 20, 115, 22, 120, 60, 70, 26, 222, 45, 80, 76, 150, 32, 240, 34, 177, 84, 100, 80, 345, 40, 110, 96, 296, 44, 300, 46, 210, 184, 130, 50, 498, 75, 225, 120, 240, 56, 380, 112, 370, 132, 160, 62, 720, 64
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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FORMULA
| a(n) is multiplicative: if the canonical factorization of n is the product of p^e(p) over primes then a(n) = product a(p^e(p)). For a prime p: a(p) = p + 3.
a(p^e) = (p^(e+2)+p^(e+1)+1+2*e-3*p-2*e*p)/(p-1)^2.
a(n) = Sum_{i|n, j|n} gcd(i, j). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 28 2001
Also a(n) = Sum_{d|n} d*tau((n/d)^2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 01 2002
Also a(n) = Sum_{d|n} phi(n/d)*tau(d)^2.
Inverse Moebius transform of A060648. [From Vladeta Jovovic (vladeta(AT)eunet.yu), Mar 31 2009]
Dirichlet g.f. zeta^3(s)*zeta(s-1)/zeta(2*s). - R. J. Mathar, Mar 14 2011
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EXAMPLE
| a(2) = 5 because for the group C_2 X C_2 there are the following subgroups: the trivial subgroup, the whole group and the three subgroups of order 2.
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MAPLE
| for n from 1 to 200 do: ans := 1: for i from 1 to nops(ifactors(n)[2]) do p := ifactors(n)[2][i][1]: e := ifactors(n)[2][i][2]: ans := ans*(p^(e+2)+p^(e+1)+1+2*e-3*p-2*e*p)/(p-1)^2: od: printf(`%d, `, ans): od:
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MATHEMATICA
| ppCase[ {p_Integer, e_Integer} ] := (1-2*e*(p-1)+p*(p^e*(1+p)-3))/(p-1)^2; Table[ Times @@ (ppCase /@ FactorInteger[ i ]), {i, 1, 100} ]
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CROSSREFS
| Cf. A060648, A050488, A054584.
Cf. A000005, A062369.
Sequence in context: A099330 A180686 A145491 * A064949 A160109 A191213
Adjacent sequences: A060721 A060722 A060723 * A060725 A060726 A060727
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KEYWORD
| nonn,mult
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AUTHOR
| Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
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EXTENSIONS
| Formula and more terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 06 2001
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