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

3,4

Comments

Csajbók and Kasza call this the tau-iteration length.

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.

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A036459, A060937, A061286, A200810.

Sequence in context: A341345 A068067 A046926 * A074398 A144765 A308062

Adjacent sequences: A200812 A200813 A200814 * A200816 A200817 A200818

Keyword

nonn

Author

Charles R Greathouse IV, Nov 23 2011

Status

approved