%I #17 Feb 27 2020 22:47:08
%S 10,15,20,25,30,35,40,45,50,51,52,53,54,55,56,57,58,60,65,69,70,75,78,
%T 80,85,87,90,95,96,98,100,101,102,103,104,105,106,107,108,109,110,115,
%U 120,125,130,135,140,145,150,151,152,153,154,155,156,157,158,159,160,165,169,170,175,178,180,185,187,190,196,200,201,202,203
%N Numbers whose waterfall sequence ends in 0,0,0,....
%C The "waterfall" sequence S for a given starting value S(1) > 9 is defined as S(n) = d(n)*d(n-1) (n > 1), where d(n) is the n-th digit of the sequence.
%C When S(1) has a digit 0 or 5, then S is likely to end up in repeating zeros. A210653 lists counterexamples, and A210614 is the subsequence of terms having no digit 0 or 5.
%H E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/Waterfalls.htm">Waterfalls (of multiplications)</a>, Mar 27 2012
%H E. Angelini, <a href="/A210614/a210614.pdf">Waterfalls (of multiplications)</a> [Cached copy, with permission]
%o (PARI) is_A210652(n,L=2e3)={ my(S=eval(Vec(Str(n)))); for(i=1,L,(S=concat(vecextract(S,"^1"),eval(Vec(Str(n=S[1]*S[2]))))) | return(1))} \\ The optional 2nd parameter L is the limit prior to which the remaining digits must be all zero. The result may be wrong if L is not large enough, but large values of L yield slow computations for values NOT in this sequence.
%o for(n=10,999, is_A210652(n) & print1(n","))
%Y Cf. A210653.
%K nonn,base
%O 1,1
%A _M. F. Hasler_, Mar 27 2012