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

0,11

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

Links

Table of n, a(n) for n=0..86.

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000045, A031286.

Sequence in context: A351014 A124767 A319443 * A266499 A226621 A112933

Adjacent sequences: A130630 A130631 A130632 * A130634 A130635 A130636

Keyword

easy,nonn,base

Author

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

Extensions

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

Status

approved