A058972 For a rational number p/q let f(p/q) = sum of aliquot divisors of p+q divided by number of divisors of p+q; sequence gives numbers k such that, starting at k/1 and iterating f, an integer is eventually reached.

3, 9, 15, 24, 25, 29, 33, 35, 50, 51, 55, 57, 59, 63, 73, 79, 80, 81, 85, 87, 89, 90, 95, 99, 105, 119, 120, 121, 128, 131, 139, 143, 145, 169, 177, 179, 181, 183, 193, 195, 201, 203, 204, 211, 215, 217, 218, 219, 221, 225, 227, 233, 247, 248, 255, 273, 275, 288

Offset

1,1

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

P. Schogt, The Wild Number Problem: math or fiction?, arXiv preprint arXiv:1211.6583 [math.HO], 2012. - From N. J. A. Sloane, Jan 03 2013

Example

f(9/1) = 8/4 = 2, an integer, so 9 is in the sequence;
f(10/1) = 1/2 and f(1/2)=1/2, so 10 is not in the sequence.

Mathematica

f[r_] := If[init == False && IntegerQ[r], r, init = False; p = Numerator[r]; q = Denominator[r]; d = Most[Divisors[p+q]]; Total[d]/(Length[d]+1)]; ok[n_] := IntegerQ[ init = True; FixedPoint[f, n/1]]; ok[1] = False; A058972 = Select[ Range[300], ok] (* Jean-François Alcover, Dec 21 2011 *)

Prog

(Haskell)
import Data.Ratio ((%), numerator, denominator)
a058972 n = a058972_list !! (n-1)
a058972_list = map numerator $ filter ((f [])) [1..] where
   f ys q = denominator y == 1 || not (y `elem` ys) && f (y : ys) y
            where y = a001065 q' % a000005 q'
                  q' = numerator q + denominator q
-- Reinhard Zumkeller, Jun 15 2013
(PARI) f2(p, q) = (sigma(p+q)-p-q)/numdiv(p+q);
f1(r) = f2(numerator(r), denominator(r));
loop(list) = {my(v=Vecrev(list)); for (i=2, #v, if (v[i] == v[1], return(1)); ); }
ff(n) = {my(ok=0, m=f2(n, 1), list=List()); while(denominator(m) != 1, m = f1(m); listput(list, m); if (loop(list), return (0)); ); return(m); }
isok(m) = ff(m) > 0; \\ Michel Marcus, Feb 09 2022

Crossrefs

Cf. A058971, A058973.

Cf. A001065, A000005.

Sequence in context: A246298 A139419 A323031 * A026222 A192720 A357167

Adjacent sequences: A058969 A058970 A058971 * A058973 A058974 A058975

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jan 14 2001

Extensions

Corrected and extended by Matthew Conroy, Apr 18 2001

Status

approved