A130631 Multiplicative persistence of Fibonacci numbers.

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, 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, 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

Offset: 0

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 19 2007

Keywords

• easy
• nonn
• base

Comments

From the 184th terms on all the Fibonacci numbers have some digits equal to zero (see A076564), thus their persistence is equal to 1.

Examples

			3524578 -> 3*5*2*4*5*7*8 = 33600 -> 3*3*6*0*0 = 0 -> persistence = 2.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A000045, A076564.

Programs

Maple

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);

Mathematica

Table[Length[NestWhileList[Times@@IntegerDigits[#]&, Fibonacci[n], #>=10&]], {n, 0, 102}]-1 (* James C. McMahon, Feb 11 2025 *)

Formula

a(n) = A031346(A000045(n)). - Michel Marcus, Feb 11 2025