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.
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
Links
- Antti Karttunen, Table of n, a(n) for n = 1..8191
- 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.
Programs
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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
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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 *) -
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
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.
Extensions
More terms from Antti Karttunen, Oct 09 2018
Comments