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A160908
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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.
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4
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1, 255, 3280, 32640, 97656, 836400, 960800, 4177920, 7173360, 24902280, 21435888, 107059200, 67977560, 245004000, 320311680, 534773760, 435984840, 1829206800, 943531280, 3187491840, 3151424000, 5466151440, 3559590240, 13703577600
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of lattices L in Z^8 such that the quotient group Z^8 / L is C_n. - Álvar Ibeas, Oct 30 2015
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(7e-7) * (p^8-1) / (p-1).
For squarefree n, a(n) = A000203(n^7). (End)
Sum_{k=1..n} a(k) ~ c * n^8, where c = (1/8) * Product_{p prime} (1 + (p^7-1)/((p-1)*p^8)) = 0.2423008904... .
Sum_{k>=1} 1/a(k) = zeta(7)*zeta(8) * Product_{p prime} (1 - 2/p^8 + 1/p^15) = 1.004270064601... . (End)
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MATHEMATICA
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f[p_, e_] := p^(7*e - 7) * (p^8-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)
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PROG
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(PARI) vector(30, n, sumdiv(n^7, d, if(ispower(d, 8), moebius(sqrtnint(d, 8))*sigma(n^7/d), 0))) \\ Altug Alkan, Oct 30 2015
(PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p^(7*f[i, 2]-7)*(p^8-1)/(p-1); f[i, 2] = 1; ); factorback(f); } \\ Michel Marcus, Nov 12 2015
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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