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

1,3

Comments

The absolute values are Pazderski's multiplicative psi(n). - R. J. Mathar, Apr 03 2012

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.

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

Adjacent sequences: A153035 A153036 A153037 * A153039 A153040 A153041

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