A059175 For a rational number p/q let f(p/q) = p*q divided by the sum of digits of p and q; a(n) is obtained by iterating f, starting at n/1, until an integer is reached, or if no integer is ever reached then a(n) = 0.

0, 66, 66, 462, 180, 66, 31395, 714, 72, 9, 5, 15, 3, 36, 42, 39, 2, 9, 45, 462, 12, 12, 90, 3703207920, 1692600, 84, 234, 27, 3043425, 74613, 6, 7930296, 264, 4290, 510, 315, 315, 73302369360, 1155, 3, 8, 239872017, 6, 4386, 1989, 18, 17740866, 499954980

Offset: 0

Author

Floor van Lamoen, Jan 15 2001

Keywords

• base
• easy
• nonn
• nice

Examples

			3/1 -> 3/4 -> 12/7 -> 84/10=42/5 -> 210/11 -> 2310/5 = 462 so a(3)=462.
84/1 -> 84/13 -> 273/4 -> 273/4 -> ...  so a(84) = 0.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..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

Crossrefs

Cf. A007953, A058971, A214866.

Programs

Haskell

import Data.Ratio ((%), numerator, denominator)
a059175 n = f [n % 1] where
   f xs@(x:_) | denominator y == 1 = numerator y
              | y `elem` xs        = 0
              | otherwise          = f (y : xs)
              where y = (numerator x * denominator x) %
                        (a007953 (numerator x) + a007953 (denominator x))
-- Reinhard Zumkeller, Mar 11 2013

Mathematica

f[Rational[p_, q_]] := p*q/(Total[ IntegerDigits[p]] + Total[ IntegerDigits[q]]); f[n_Integer] := n/(1 + Total[ IntegerDigits[n]]); a[n_] := If[ IntegerQ[ r = NestWhile[f, n, Not[#1 == #2 || #1 != #2 && IntegerQ[#2]]&, 2]], r, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 03 2013 *)

PARI

f2(p,q) = p*q/(sumdigits(p)+sumdigits(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)););}
a(n) = {if (n==0, return(0)); 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);} \\ Michel Marcus, Feb 09 2022

Formula

a(A214866(n)) = 0. - Reinhard Zumkeller, Mar 11 2013

Extensions

Corrected and extended by Naohiro Nomoto, Jul 20 2001