%I #23 Jun 29 2016 06:43:08
%S 1,2,3,4,5,18,7,8,9,20,11,48,13,112,45,16,17,468,19,480,21,88,23,72,
%T 25,52,27,196,29,180,31,32,99,68,35,36,37,152,39,80,41,1344,43,176,
%U 810,368,47,192,49,50,459,104,53,162,55,448,57,116,59,9360,61,1984,63,64,65
%N Least number m such that n is the denominator of sigma_{-1}(m), or zero if no such exists.
%C First occurrence of n in A017666.
%C Conjecture: a(n) is never zero. Checking up to 1000000, the smallest number not found is 210; and a(210) = 26611200.
%C n|a(n), since sigma_{-1}(n) = sigma(n)/n. a(n) = n for n any prime power (and many others).
%C Up to 1000, the maximum value is a(330) = 1890345600. - _Michel Marcus_, Aug 14 2012
%C Actually, a(n) = n, for n in A014567. - _Michel Marcus_, Dec 28 2013
%C Up to 10000, the largest term is a(9570) = 22033432080000. - _Giovanni Resta_, Mar 22 2014
%H Michel Marcus and Giovanni Resta, <a href="/A162657/b162657.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Michel Marcus)
%t a[n_] := Catch[ For[ lim = Quotient[2*10^9, n]*n; k = 0, k <= lim, k = k + n, If[Denominator[ DivisorSigma[-1, k]] == n, Throw[k]]; If[k >= lim, Throw[0]]]]; a[1]=1; Table[ an = a[n]; Print[{n, an}]; an , {n, 1, 1000}] (* _Jean-François Alcover_, Aug 14 2012 *)
%o (PARI) al(n,lim=100000)=local(r,d);r=vector(n);for(k=1,lim,d=denominator(sigma(k,-1));if(d<=n&&r[d]==0,r[d]=k));r
%o a(n,lim=1000000)=forstep(m=n,lim,n,if(denominator(sigma(m,-1))==n,return(m)));0
%Y Cf. A017666, A017665, A000203.
%K nonn
%O 1,2
%A _Franklin T. Adams-Watters_, Jul 08 2009