A200815 Number of iterations of k -> d(k) until n reaches an odd prime.

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, 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, 0, 4, 0, 2, 3, 3, 2, 3, 0, 3, 1, 2, 0, 4, 2, 2, 2, 3, 0

Offset: 3

Author

Charles R Greathouse IV, Nov 23 2011

Keywords

• nonn

Comments

Csajbók and Kasza call this the tau-iteration length.

Examples

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

Links

• Antti Karttunen, Table of n, a(n) for n = 3..10000
• Tímea Csajbók and János Kasza, Iterating the tau-function, Annales Univ. Sci. Budapest., Sec. Math. 35 (2011), pp. 83-93.

Crossrefs

Cf. A036459, A060937, A061286, A200810.

Programs

Mathematica

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

PARI

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

Formula

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

a(n) = A036459(n)-1 = A060937(n)-2, for n >= 3. - Antti Karttunen, Oct 06 2017