A080736 Multiplicative function defined by a(1)=1, a(2)=0, a(2^r) = phi(2^r) (r>1), a(p^r) = phi(p^r) (p odd prime, r>=1), where phi is Euler's function A000010.

1, 0, 2, 2, 4, 0, 6, 4, 6, 0, 10, 4, 12, 0, 8, 8, 16, 0, 18, 8, 12, 0, 22, 8, 20, 0, 18, 12, 28, 0, 30, 16, 20, 0, 24, 12, 36, 0, 24, 16, 40, 0, 42, 20, 24, 0, 46, 16, 42, 0, 32, 24, 52, 0, 40, 24, 36, 0, 58, 16, 60, 0, 36, 32, 48, 0, 66, 32, 44, 0, 70, 24, 72, 0, 40, 36, 60, 0, 78, 32

Offset

1,3

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(A016825(n)) = 0; a(A000040(n)) = A000040(n) - 1. - Reinhard Zumkeller, Jun 11 2012

a(n) = if n mod 4 = 2 then 0 else A000010(n). - Reinhard Zumkeller, Jun 13 2012

From Amiram Eldar, Nov 02 2023: (Start)

Multiplicative with a(2) = 0, a(2^e) = 2^(e-1) for e >= 2, and a(p^e) = (p-1)*p^(e-1) for an odd prime p.

Dirichlet g.f.: (1 - 2^(1-s) + 1/(2^s-1)) * zeta(s-1) / zeta(s).

Sum_{k=1..n} a(k) ~ (5/(2*Pi^2)) * n^2. (End)

Mathematica

a[n_] := If[Mod[n, 4] == 2, 0, EulerPhi[n]]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)

Prog

(PARI) {for(n=1, 81, f=factor(n); print1(if(n==1, 1, if(f[1, 1]==2&&f[1, 2]==1, 0, prod(j=1, matsize(f)[1], eulerphi(f[j, 1]^f[j, 2])))), ", "))}
(Haskell)
a080736 n = if n `mod` 4 == 2 then 0 else a000010 n
-- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012

Crossrefs

Cf. A080737.

Cf. A000010, A141809.

Sequence in context: A030207 A366561 A061006 * A326127 A276151 A144412

Adjacent sequences: A080733 A080734 A080735 * A080737 A080738 A080739

Keyword

nonn,easy,mult

Author

N. J. A. Sloane, Mar 08 2003

Extensions

More terms and PARI code from Klaus Brockhaus, Mar 10 2003

Status

approved