A166633 Totally multiplicative sequence with a(p) = 3*(p-1) for prime p.

1, 3, 6, 9, 12, 18, 18, 27, 36, 36, 30, 54, 36, 54, 72, 81, 48, 108, 54, 108, 108, 90, 66, 162, 144, 108, 216, 162, 84, 216, 90, 243, 180, 144, 216, 324, 108, 162, 216, 324, 120, 324, 126, 270, 432, 198, 138, 486, 324, 432

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

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).

a(n) = A165824(n) * A003958(n) = 3^bigomega(n) * A003958(n) = 3^A001222(n) * A003958(n).

Mathematica

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]*3^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (3*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

Prog

(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

Crossrefs

Cf. A001222, A003958, A165824, A166634.

Sequence in context: A271449 A261956 A344683 * A310154 A253277 A310155

Adjacent sequences: A166630 A166631 A166632 * A166634 A166635 A166636

Keyword

nonn,easy,mult

Author

Jaroslav Krizek, Oct 18 2009

Status

approved