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A054584
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Number of subgroups of the group generated by a^n=1, b^3=1 and ab=ba.
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2
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2, 4, 6, 6, 4, 12, 4, 8, 10, 8, 4, 18, 4, 8, 12, 10, 4, 20, 4, 12, 12, 8, 4, 24, 6, 8, 14, 12, 4, 24, 4, 12, 12, 8, 8, 30, 4, 8, 12, 16, 4, 24, 4, 12, 20, 8, 4, 30, 6, 12, 12, 12, 4, 28, 8, 16, 12, 8, 4, 36, 4, 8, 20, 14, 8, 24, 4, 12, 12, 16, 4, 40, 4, 8, 18, 12, 8, 24, 4, 20, 18, 8, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Also the number of subgroups of the group C_n X C_3 (where C_n is the cyclic group of order n). Number of subgroups of the group C_n X C_m is Sum_{i|n,j|m} gcd(i,j).
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FORMULA
| a(n) = tau(n)+3*tau(n/3)+A035191(n) if n is congruent to 0 mod 3 else tau(n)+A035191(n), where A035191(n) is the number of divisors of n that are not congruent to 0 mod 3.
a(n)/2 is multiplicative with a(3^e)=2e+1 and a(p^e)=e+1 for p<>3.
Moebius transform is period 3 sequence [2, 2, 4, ...]. - Michael Somos Sep 20 2005
G.f.: Sum_{k>0} x^k(2+2*x^k+4*x^(2k))/(1-x^(3k)).
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MAPLE
| for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=3 then b := 2*e+1:else b := e+1:fi:s := s*b:od:printf(`%d, `, 2*s); od:
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MATHEMATICA
| f[d_ /; Mod[d, 3] == 0] = 4; f[_] = 2; a[n_] := Total[f /@ Divisors[n]]; Table[a[n], {n, 1, 100}](* From Jean-François Alcover, Nov 21 2011, after Michael Somos *)
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PROG
| (PARI) a(n)=if(n<1, 0, sumdiv(n, d, (d%3==0)*2+2)) /* Michael Somos Sep 20 2005 */
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CROSSREFS
| Cf. A060710, A060724, A062011, A060648, A035191, A000005.
Sequence in context: A010587 A134920 A011031 * A049041 A092337 A050823
Adjacent sequences: A054581 A054582 A054583 * A054585 A054586 A054587
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KEYWORD
| nonn,easy,nice
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), Apr 12 2000
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EXTENSIONS
| Additional comments from Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 25 2001
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