A060937 - OEIS

%I #17 Jan 05 2025 19:51:36

%S 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,

%T 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,

%U 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

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

%C 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.

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

%H Charles R Greathouse IV, <a href="/A060937/b060937.txt">Table of n, a(n) for n = 2..10000</a>

%H Paul Erdős and Imre Kátai, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-3/erdos.pdf">On the growth of d_k(n)</a>, Fibonacci Quarterly, Vol. 7, No. 3 (1969), pp. 267-274.

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

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

%p 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;

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

%o (PARI) a(n)=my(t=1);while(n>2,n=numdiv(n);t++);t \\ _Charles R Greathouse IV_, Apr 07 2012

%Y Equals A036459 + 1.

%Y Cf. A000005, A049108, A003434.

%K nonn

%O 2,2

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

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