A202535 - OEIS

A202535

a(n) = n*phi(n)*abs( mobius(n) ).

1, 2, 6, 0, 20, 12, 42, 0, 0, 40, 110, 0, 156, 84, 120, 0, 272, 0, 342, 0, 252, 220, 506, 0, 0, 312, 0, 0, 812, 240, 930, 0, 660, 544, 840, 0, 1332, 684, 936, 0, 1640, 504, 1806, 0, 0, 1012, 2162, 0, 0, 0, 1632, 0, 2756, 0, 2200, 0, 2052, 1624, 3422

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OFFSET

1,2

COMMENTS

The inverse Mobius transform is b(n>=1) = 1, 3, 7, 3, 21, 21, 43, 3,7, 63, 11, 21,...., multiplicative with b(p^e) = A002061(p), e>=1 (see A119959). - Mathar

a(n) > 0 only when n is squarefree. - Alonso del Arte, Dec 20 2011

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A002618(n) *A008966(n).

Multiplicative with a(p^e) = (p-1)*p if e=1, a(p^e)=0 if e>1.

Dirichlet g.f.: Sum_(n>=1) a(n)/n^s = Product_{primes p} (1-p^(1-s)+p^(2-s)).

From Vaclav Kotesovec, Jun 24 2020: (Start)

Dirichlet g.f.: zeta(s-2)*Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = A065464/3 = 0.142749835... (End)

EXAMPLE

a(5) = 20 because 5 phi(5) |mu(5)| = 5 * 4 * |(-1)| = 20.

MATHEMATICA

Table[n EulerPhi[n] Abs[MoebiusMu[n]], {n, 60}] (* Alonso del Arte, Dec 20 2011 *)

f[p_, e_] := If[e == 1, (p-1)*p, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

PROG

(PARI) a(n)=n*eulerphi(n)*abs(moebius(n)) \\ Charles R Greathouse IV, Dec 20 2011

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 24 2020

CROSSREFS

Cf. A079579.

Sequence in context: A243015 A139717 A285119 * A138703 A106458 A354351

Adjacent sequences: A202532 A202533 A202534 * A202536 A202537 A202538

KEYWORD

nonn,mult,easy

AUTHOR

R. J. Mathar, Dec 20 2011

STATUS

approved