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A344305
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Number of cyclic subgroups of the group (C_n)^9, where C_n is the cyclic group of order n.
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5
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1, 512, 9842, 131328, 488282, 5039104, 6725602, 33620224, 64576643, 250000384, 235794770, 1292530176, 883708282, 3443508224, 4805671444, 8606777600, 7411742282, 33063241216, 17927094322, 64125098496, 66193374884, 120726922240, 81870575522, 330890244608
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OFFSET
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1,2
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COMMENTS
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Inverse Moebius transform of A160953.
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LINKS
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FORMULA
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a(n) = Sum_{x_1|n, x_2|n, ..., x_9|n} phi(x_1)*phi(x_2)* ... *phi(x_9)/phi(lcm(x_1, x_2, ..., x_9)).
If p is prime, a(p) = 1 + (p^9 - 1)/(p - 1).
Multiplicative with a(p^e) = 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)).
Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} ((1-1/p^8)/(p^2*(1-1/p))) = 0.2161023934... . (End)
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MATHEMATICA
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f[p_, e_] := 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)
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PROG
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(PARI) a160953(n) = sumdiv(n, d, moebius(n/d)*d^9)/eulerphi(n);
a(n) = sumdiv(n, d, a160953(d));
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CROSSREFS
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Cf. A000010, A013667, A160953, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344306.
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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