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A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k). 46
1, 1, 2, 1, 4, 2, 6, 1, 4, 4, 10, 2, 12, 6, 8, 1, 16, 4, 18, 4, 12, 10, 22, 2, 16, 12, 8, 6, 28, 8, 30, 1, 20, 16, 24, 4, 36, 18, 24, 4, 40, 12, 42, 10, 16, 22, 46, 2, 36, 16, 32, 12, 52, 8, 40, 6, 36, 28, 58, 8, 60, 30, 24, 1, 48, 20, 66, 16, 44, 24, 70, 4, 72, 36, 32, 18, 60, 24, 78, 4, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Completely multiplicative.

a(n) = A000010(n) iff n is squarefree (see A005117). - Reinhard Zumkeller, Nov 05 2004

Dirichlet inverse of A097945. - R. J. Mathar, Aug 29 2011

a(n) = abs(A125131(n)). - Tom Edgar, May 26 2014

LINKS

T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)

Index to divisibility sequences

FORMULA

Multiplicative with a(p^e) = (p-1)^e. - David W. Wilson, Aug 01 2001

MAPLE

a:= n-> mul((i[1]-1)^i[2], i=ifactors(n)[2]):

seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

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]; Table[ DirichletInverse[ muphi][n], {n, 1, 81}] (* Jean-Fran├žois Alcover, Dec 12 2011, after R. J. Mathar *)

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 10 2016 *)

PROG

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-p*X+X))[n]) /* Ralf Stephan */

(Haskell)

a003958 1 = 1

a003958 n = product $ map (subtract 1) $ a027746_row n

-- Reinhard Zumkeller, Apr 09 2012, Mar 02 2012

CROSSREFS

Cf. A003959, A168065, A168066, A027746, A006093, A027748, A124010.

Sequence in context: A187203 A187202 A125131 * A326140 A082729 A326069

Adjacent sequences:  A003955 A003956 A003957 * A003959 A003960 A003961

KEYWORD

nonn,mult,nice

AUTHOR

Marc LeBrun

EXTENSIONS

Definition reedited (from formula) by Daniel Forgues, Nov 17 2009

STATUS

approved

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Last modified October 19 15:46 EDT 2019. Contains 328223 sequences. (Running on oeis4.)