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A070562
Fecundity of n.
8
0, 10, 9, 9, 8, 1, 8, 7, 7, 6, 0, 8, 7, 7, 6, 1, 6, 6, 5, 3, 0, 5, 5, 4, 5, 2, 4, 5, 2, 3, 0, 3, 4, 2, 2, 1, 3, 3, 3, 2, 0, 4, 1, 2, 1, 3, 1, 2, 1, 4, 0, 5, 3, 8, 2, 1, 4, 2, 2, 1, 0, 2, 2, 5, 5, 2, 1, 1, 7, 5, 0, 4, 4, 2, 1, 1, 6, 5, 3, 2, 0, 4, 2, 1, 7, 3, 3, 3, 4, 3, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0
OFFSET
0,2
COMMENTS
Start with x=n, repeatedly replace x by x + product of digits of x until the product is 0; fecundity = number of steps. a(0) = 0 by convention.
REFERENCES
P. Tougne, Jeux Mathematiques column, Pour La Science (French edition of "Scientific American"), Vol. 82, Aug. 1984, Prob. 6, pp. 101, 104.
EXAMPLE
1 -> 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 -> 102 has fecundity 10.
MATHEMATICA
f[ n_ ] := Block[ {a=n, b, c=0}, While[ b=Times@@IntegerDigits[ a ]; b>0, a=a+b; c++ ]; c ]; f[ 0 ]=0; Table[ f[ n ], {n, 0, 100} ]
f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 2; Array[f, 105, 0] (* Robert G. Wilson v, Jun 27 2010 *)
PROG
(PARI) prodig(n) = local(s, d); if(n==0, s=0, s=1; while(n>0, d=divrem(n, 10); n=d[1 ]; s=s*d[2 ])); s for(n=0, 92, x=n; c=0; while((d=prodig(x))!=0, c++; x=x+d); print1(c, ", "))
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
N. J. A. Sloane, May 07 2002
EXTENSIONS
Edited and extended by Klaus Brockhaus, May 08 2002
Clarified the definition of fecundity and improved the Mathematica program. - T. D. Noe, Oct 06 2008
More terms from Robert G. Wilson v, Jun 27 2010
STATUS
approved