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%I #15 Oct 17 2023 05:42:28
%S 1,3,6,9,12,18,18,27,36,36,30,54,36,54,72,81,48,108,54,108,108,90,66,
%T 162,144,108,216,162,84,216,90,243,180,144,216,324,108,162,216,324,
%U 120,324,126,270,432,198,138,486,324,432
%N Totally multiplicative sequence with a(p) = 3*(p-1) for prime p.
%H G. C. Greubel, <a href="/A166633/b166633.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = (3*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)-1)^e(k).
%F a(n) = A165824(n) * A003958(n) = 3^bigomega(n) * A003958(n) = 3^A001222(n) * A003958(n).
%t DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] :=
%t 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]*3^(PrimeOmega[m]), {m, 1, 100}] (* _G. C. Greubel_, May 20 2016 *)
%t f[p_, e_] := (3*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 17 2023 *)
%o (PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = 3*(f[k,1]-1)); factorback(f);} \\ _Michel Marcus_, May 20 2016
%Y Cf. A001222, A003958, A165824, A166634.
%K nonn,easy,mult
%O 1,2
%A _Jaroslav Krizek_, Oct 18 2009
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