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A153038 Denominators of the fixed point a=(a_1,a_2,...) of the transformation x'= fix(x) where fix(x)n = Sum{d|n} d x_d, i.e., fix(a)=a.

%I A153038 #27 Oct 09 2018 15:11:57
%S A153038 1,-1,-2,3,-4,2,-6,-21,16,4,-10,-6,-12,6,8,315,-16,-16,-18,-12,12,10,
%T A153038 -22,42,96,12,-416,-18,-28,-8,-30,-9765,20,16,24,48,-36,18,24,84,-40,
%U A153038 -12,-42,-30,-64,22,-46,-630,288,-96,32,-36,-52,416,40,126,36,28,-58,24,-60,30,-96,615195,48,-20,-66,-48,44,-24,-70,-336,-72,36
%N A153038 Denominators of the fixed point a=(a_1,a_2,...) of the transformation x'= fix(x) where fix(x)_n = Sum_{d|n} d x_d, i.e., fix(a)=a.
%C A153038 The absolute values are Pazderski's multiplicative psi(n). - _R. J. Mathar_, Apr 03 2012
%H A153038 Antti Karttunen, <a href="/A153038/b153038.txt">Table of n, a(n) for n = 1..8191</a>
%H A153038 M. Baake and N. Neumaerker, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Baake/baake7.html">A note on the relation between fixed point and orbit count sequences</a>, Journal of Integer Sequences (2009) 09.4.4.
%H A153038 G. Pazderski, <a href="http://dx.doi.org/10.1007/BF01240807">Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehoren</a>, Archiv math. 10 (1) (1959) 331.
%H A153038 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, Journal of Integer Sequences, 4 (2001), article 01.2.1.
%F A153038 For n with prime factorization n = p_1^{r_1}*...*p_s^{r_s} the n-th term is a(n) = Product_{k=1..s} Product_{j=1..r_k} (1 - p_k^j).
%F A153038 G.f.: The Dirichlet series for 1/a(n) is Product_{j>= 1} 1/zeta(s+j) = Product_{p prime} Product_{j>= 1} (1 - 1/p^(s+j)) where zeta(s) is Riemann's zeta function.
%p A153038 A153038 := proc(n)
%p A153038         local f,a,p,e;
%p A153038         if n = 1 then
%p A153038                 1;
%p A153038         else
%p A153038                 a := 1 ;
%p A153038                 for f in ifactors(n)[2] do
%p A153038                         p := op(1,f) ;
%p A153038                         e := op(2,f) ;
%p A153038                         a := a*mul(1-p^s,s=1..e) ;
%p A153038                 end do:
%p A153038                 return a ;
%p A153038         end if;
%p A153038 end proc: # _R. J. Mathar_, Apr 03 2012
%t A153038 a[1] = 1; a[n_] := (x = 1; Do[p = f[[1]]; e = f[[2]]; x = x*Product[1 - p^s, {s, 1, e}], {f, FactorInteger[n]}]; x); Table[a[n], {n, 1, 46}] (* _Jean-François Alcover_, May 15 2012, after _R. J. Mathar_ *)
%o A153038 (PARI) a(n)=my(f=factor(n));prod(k=1,#f[,1],prod(j=1,f[k,2], 1-f[k,1]^j)) \\ _Charles R Greathouse IV_, Sep 18 2012
%K A153038 easy,eigen,frac,mult,sign
%O A153038 1,3
%A A153038 Natascha Neumaerker (naneumae(AT)math.uni-bielefeld.de), Dec 17 2008
%E A153038 More terms from _Antti Karttunen_, Oct 09 2018