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A166640
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Totally multiplicative sequence with a(p) = 9*(p-1) for prime p.
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1
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1, 9, 18, 81, 36, 162, 54, 729, 324, 324, 90, 1458, 108, 486, 648, 6561, 144, 2916, 162, 2916, 972, 810, 198, 13122, 1296, 972, 5832, 4374, 252, 5832, 270, 59049, 1620, 1296, 1944, 26244, 324, 1458, 1944, 26244, 360, 8748, 378, 7290, 11664, 1782, 414, 118098
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (9*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (9*(p(k)-1)^e(k).
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MATHEMATICA
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DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] :=
DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*9^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (9*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = 9*(f[k, 1]-1)); factorback(f); } \\ Michel Marcus, May 21 2016
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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