A130633 Additive persistence of Fibonacci numbers.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 2

Offset: 0

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 19 2007, corrected Jun 22 2007

Keywords

• easy
• nonn
• base

Comments

Up to 10000 the maximum value is 4.

Up to 10^22 the maximum value is 4. Conjecture: for n > 15, a(n) > 1. I checked the conjecture up to 10^10. - Charles R Greathouse IV, Feb 12 2025

Examples

			3524578 -> 3+5+2+4+5+7+8 = 34 -> 3+4 = 7 -> persistence = 2.
		

Crossrefs

Cf. A000045, A031286.

Programs

Maple

with(numtheory): with(combinat): P:=proc(n) local a,t; t:=0; a:=fibonacci(n); while a>9 do t:=t+1; a:=convert(convert(a,base,10),`+`); od; t;
end: seq(P(i),i=0..10^2);

Mathematica

Table[Length[NestWhileList[Plus@@IntegerDigits[#]&, Fibonacci[n], #>=10&]], {n, 0, 86}]-1 (* James C. McMahon, Feb 11 2025 *)

PARI

ap(n)=my(s); while(n>9, n=sumdigits(n); s++); s
a(n)=ap(fibonacci(n)) \\ Charles R Greathouse IV, Feb 12 2025

Formula

a(n) = A031286(A000045(n)). - Michel Marcus, Feb 12 2025

Extensions

Corrected entries and changed Maple code by Paolo P. Lava, Dec 19 2017