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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.
4
1, -1, -2, 3, -4, 2, -6, -21, 16, 4, -10, -6, -12, 6, 8, 315, -16, -16, -18, -12, 12, 10, -22, 42, 96, 12, -416, -18, -28, -8, -30, -9765, 20, 16, 24, 48, -36, 18, 24, 84, -40, -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
OFFSET
1,3
COMMENTS
The absolute values are Pazderski's multiplicative psi(n). - R. J. Mathar, Apr 03 2012
LINKS
M. Baake and N. Neumaerker, A note on the relation between fixed point and orbit count sequences, Journal of Integer Sequences (2009) 09.4.4.
G. Pazderski, Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehoren, Archiv math. 10 (1) (1959) 331.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, Journal of Integer Sequences, 4 (2001), article 01.2.1.
FORMULA
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).
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.
MAPLE
A153038 := proc(n)
local f, a, p, e;
if n = 1 then
1;
else
a := 1 ;
for f in ifactors(n)[2] do
p := op(1, f) ;
e := op(2, f) ;
a := a*mul(1-p^s, s=1..e) ;
end do:
return a ;
end if;
end proc: # R. J. Mathar, Apr 03 2012
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A193024 A340368 A367175 * A368698 A324911 A220335
KEYWORD
easy,eigen,frac,mult,sign
AUTHOR
Natascha Neumaerker (naneumae(AT)math.uni-bielefeld.de), Dec 17 2008
EXTENSIONS
More terms from Antti Karttunen, Oct 09 2018
STATUS
approved