A166640 Totally multiplicative sequence with a(p) = 9*(p-1) for prime p.

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

Offset: 1

Author

Jaroslav Krizek, Oct 18 2009

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A001222, A003958, A165830.

Programs

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]*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 *)

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

Formula

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).

a(n) = A165830(n) * A003958(n) = 9^bigomega(n) * A003958(n) = 9^A001222(n) * A003958(n).