A060937 Number of iterations of d(n) (A000005) needed to reach 2 starting at n (n is counted).

1, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 3, 2, 5, 2, 5, 4, 4, 2, 5, 3, 4, 4, 5, 2, 5, 2, 5, 4, 4, 4, 4, 2, 4, 4, 5, 2, 5, 2, 5, 5, 4, 2, 5, 3, 5, 4, 5, 2, 5, 4, 5, 4, 4, 2, 6, 2, 4, 5, 3, 4, 5, 2, 5, 4, 5, 2, 6, 2, 4, 5, 5, 4, 5, 2, 5, 3, 4, 2, 6, 4, 4, 4, 5, 2, 6, 4, 5, 4, 4, 4, 6, 2, 5, 5, 4, 2, 5, 2, 5, 5, 4

Offset: 2

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001

Keywords

• nonn

Comments

By the definition of a(n) we have for n >= 3 the recursion a(n) = a(d(n)) + 1. a(n) = 2 iff n is an odd prime.

Examples

			If n=12 the trajectory is {12,6,4,3,2}. Its length is 5, thus a(12)=5.
		

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter 2, page 66.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
• Paul Erdős and Imre Kátai, On the growth of d_k(n), Fibonacci Quarterly, Vol. 7, No. 3 (1969), pp. 267-274.

Crossrefs

Equals A036459 + 1.

Cf. A000005, A049108, A003434.

Programs

Maple

with(numtheory): interface(quiet=true): for n from 2 to 200 do if (1=1) then temp := n: count := 1: end if; while (temp > 2) do temp := tau(temp): count := count + 1: od; printf("%d,", count); od;

Mathematica

a[n_] := -1 + Length @ FixedPointList[DivisorSigma[0, #] &, n]; Array[a, 100, 2] (* Amiram Eldar, Jul 10 2021 *)

PARI

a(n)=my(t=1);while(n>2,n=numdiv(n);t++);t \\ Charles R Greathouse IV, Apr 07 2012

Formula

0 < lim sup_{n->oo} (a(n)-1)/log(log(log(n))) < oo (Erdős and Kátai, 1969). - Amiram Eldar, Jul 10 2021

Extensions

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), May 21 2001