A162657 Least number m such that n is the denominator of sigma_{-1}(m), or zero if no such exists.

1, 2, 3, 4, 5, 18, 7, 8, 9, 20, 11, 48, 13, 112, 45, 16, 17, 468, 19, 480, 21, 88, 23, 72, 25, 52, 27, 196, 29, 180, 31, 32, 99, 68, 35, 36, 37, 152, 39, 80, 41, 1344, 43, 176, 810, 368, 47, 192, 49, 50, 459, 104, 53, 162, 55, 448, 57, 116, 59, 9360, 61, 1984, 63, 64, 65

Offset: 1

Author

Franklin T. Adams-Watters, Jul 08 2009

Keywords

• nonn

Comments

First occurrence of n in A017666.

Conjecture: a(n) is never zero. Checking up to 1000000, the smallest number not found is 210; and a(210) = 26611200.

n|a(n), since sigma_{-1}(n) = sigma(n)/n. a(n) = n for n any prime power (and many others).

Up to 1000, the maximum value is a(330) = 1890345600. - Michel Marcus, Aug 14 2012

Actually, a(n) = n, for n in A014567. - Michel Marcus, Dec 28 2013

Up to 10000, the largest term is a(9570) = 22033432080000. - Giovanni Resta, Mar 22 2014

Links

• Michel Marcus and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

Crossrefs

Cf. A017666, A017665, A000203.

Programs

Mathematica

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 *)

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
a(n,lim=1000000)=forstep(m=n,lim,n,if(denominator(sigma(m,-1))==n,return(m)));0