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 A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k). 46

%I

%S 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,

%T 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,

%U 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

%N If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).

%C Completely multiplicative.

%C a(n) = A000010(n) iff n is squarefree (see A005117). - _Reinhard Zumkeller_, Nov 05 2004

%C Dirichlet inverse of A097945. - _R. J. Mathar_, Aug 29 2011

%C a(n) = abs(A125131(n)). - _Tom Edgar_, May 26 2014

%H T. D. Noe and Daniel Forgues, <a href="/A003958/b003958.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 terms from T. D. Noe)

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F Multiplicative with a(p^e) = (p-1)^e. - _David W. Wilson_, Aug 01 2001

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

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Sep 13 2017

%t 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-François Alcover_, Dec 12 2011, after _R. J. Mathar_ *)

%t 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 *)

%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X+X))[n]) /* _Ralf Stephan_ */

%o a003958 1 = 1

%o a003958 n = product \$ map (subtract 1) \$ a027746_row n

%o -- _Reinhard Zumkeller_, Apr 09 2012, Mar 02 2012

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

%K nonn,mult,nice

%O 1,3

%A _Marc LeBrun_

%E Definition reedited (from formula) by _Daniel Forgues_, Nov 17 2009

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Last modified October 18 13:31 EDT 2019. Contains 328161 sequences. (Running on oeis4.)