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A130631 Multiplicative persistence of Fibonacci numbers.

%I A130631 #13 Feb 13 2025 12:21:13
%S A130631 0,0,0,0,0,0,0,1,1,2,3,3,2,2,4,1,2,3,2,2,2,1,4,2,3,1,3,3,4,2,1,3,1,2,
%T A130631 1,2,1,2,1,2,1,1,3,3,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,
%U A130631 1,1,1,1,1,1,1,1,1,1,3,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1
%N A130631 Multiplicative persistence of Fibonacci numbers.
%C A130631 From the 184th terms on all the Fibonacci numbers have some digits equal to zero (see A076564), thus their persistence is equal to 1.
%H A130631 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A130631 a(n) = A031346(A000045(n)). - _Michel Marcus_, Feb 11 2025
%e A130631 3524578 -> 3*5*2*4*5*7*8 = 33600 -> 3*3*6*0*0 = 0 -> persistence = 2.
%p A130631 P:=proc(n)local f0,f1,f2,i,k,w,ok,cont; f0:=0; f1:=1; print(0); print(0); for i from 0 by 1 to n do f2:=f1+f0; f0:=f1; f1:=f2; w:=1; ok:=1; k:=f2; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);
%t A130631 Table[Length[NestWhileList[Times@@IntegerDigits[#]&, Fibonacci[n], #>=10&]], {n, 0, 102}]-1 (* _James C. McMahon_, Feb 11 2025 *)
%Y A130631 Cf. A000045, A076564.
%K A130631 easy,nonn,base
%O A130631 0,10
%A A130631 _Paolo P. Lava_ and _Giorgio Balzarotti_, Jun 19 2007