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A160953
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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.
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4
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1, 511, 9841, 130816, 488281, 5028751, 6725601, 33488896, 64566801, 249511591, 235794769, 1287360256, 883708281, 3436782111, 4805173321, 8573157376, 7411742281, 32993635311, 17927094321, 63874967296, 66186639441
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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^9 such that the quotient group Z^9 / L is C_n. - Álvar Ibeas, Nov 03 2015
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(8e-8) * (p^9-1) / (p-1).
For squarefree n, a(n) = A000203(n^8). (End)
Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 + (p^8-1)/((p-1)*p^9)) = 0.2156692448... .
Sum_{k>=1} 1/a(k) = zeta(8)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^17) = 1.002068659133... . (End)
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MAPLE
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add(numtheory[mobius](n/d)*d^9, d=numtheory[divisors](n)) ;
%/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
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MATHEMATICA
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JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n];
A160953[n_]:=JordanTotient[n, 9]/JordanTotient[n];
f[p_, e_] := p^(8*e - 8) * (p^9-1) / (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(100, n, sumdiv(n^8, d, if(ispower(d, 9), moebius(sqrtnint(d, 9))*sigma(n^8/d), 0))) \\ Altug Alkan, Nov 05 2015
(PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p^(8*f[i, 2]-8)*(p^9-1)/(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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