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

%I A060937 #17 Jan 05 2025 19:51:36
%S A060937 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 A060937 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 A060937 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 A060937 Number of iterations of d(n) (A000005) needed to reach 2 starting at n (n is counted).
%C A060937 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 A060937 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter 2, page 66.
%H A060937 Charles R Greathouse IV, <a href="/A060937/b060937.txt">Table of n, a(n) for n = 2..10000</a>
%H A060937 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 A060937 0 < lim sup_{n->oo} (a(n)-1)/log(log(log(n))) < oo (Erdős and Kátai, 1969). - _Amiram Eldar_, Jul 10 2021
%e A060937 If n=12 the trajectory is {12,6,4,3,2}. Its length is 5, thus a(12)=5.
%p A060937 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 A060937 a[n_] := -1 + Length @ FixedPointList[DivisorSigma[0, #] &, n]; Array[a, 100, 2] (* _Amiram Eldar_, Jul 10 2021 *)
%o A060937 (PARI) a(n)=my(t=1);while(n>2,n=numdiv(n);t++);t \\ _Charles R Greathouse IV_, Apr 07 2012
%Y A060937 Equals A036459 + 1.
%Y A060937 Cf. A000005, A049108, A003434.
%K A060937 nonn
%O A060937 2,2
%A A060937 Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
%E A060937 More terms from Winston C. Yang (winston(AT)cs.wisc.edu), May 21 2001