A200815 - OEIS

%I #22 Nov 14 2020 17:53:12

%S 0,1,0,2,0,2,1,2,0,3,0,2,2,1,0,3,0,3,2,2,0,3,1,2,2,3,0,3,0,3,2,2,2,2,

%T 0,2,2,3,0,3,0,3,3,2,0,3,1,3,2,3,0,3,2,3,2,2,0,4,0,2,3,1,2,3,0,3,2,3,

%U 0,4,0,2,3,3,2,3,0,3,1,2,0,4,2,2,2,3,0

%N Number of iterations of k -> d(k) until n reaches an odd prime.

%C Csajbók and Kasza call this the tau-iteration length.

%H Antti Karttunen, <a href="/A200815/b200815.txt">Table of n, a(n) for n = 3..10000</a>

%H Tímea Csajbók and János Kasza, <a href="http://ac.inf.elte.hu/Vol_035_2011/083.pdf">Iterating the tau-function</a>, Annales Univ. Sci. Budapest., Sec. Math. 35 (2011), pp. 83-93.

%F a(n) <= pi(log_2(n)) = A000720(A000523(n)).

%F a(n) = A036459(n)-1 = A060937(n)-2, for n >= 3. - _Antti Karttunen_, Oct 06 2017

%e d(10) = 4 and d(4) = 3, an odd prime, so a(10) = 2.

%t nop[n_]:=Length[NestWhileList[DivisorSigma[0,#]&,n,#<3 || CompositeQ[ #]&]]-1; Array[ nop,100,3] (* _Harvey P. Dale_, Nov 14 2020 *)

%o (PARI) a(n)=my(i);while(!isprime(n),i++;n=numdiv(n));i

%Y Cf. A036459, A060937, A061286, A200810.

%K nonn

%O 3,4

%A _Charles R Greathouse IV_, Nov 23 2011