A096309 a(1)=1; for n > 1, a(n) is the number of levels in the "stacked" prime number factorization of n (prime number factorization of the exponents if necessary and so on ...).

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2

Offset: 1

Author

Franz Vrabec, Jun 27 2004

Keywords

• easy
• nonn

Comments

For n > 1: a(n)=1 iff n squarefree.

Sequence A185102 is a (better?) variant, identical except for A185102(1)=0. - M. F. Hasler, Nov 21 2013

Examples

			a(4)=2 because 4=2^2; a(8)=2 because 8=2^3; a(16)=3 because 16=2^(2^2).
a(65536) = a(2^2^2^2) = a(2^^4) = 4 is the first term larger than 3; the index of the first a(n) > 4, n = 2^^5, has 19729 digits. - _M. F. Hasler_, Nov 21 2013
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A087049, A185102.

Programs

Mathematica

f[n_Integer] := FactorInteger[n][[All, 2]]; a[n_] := Depth[f[n] //. k_Integer /; k > 1 :> f[k]] - 1; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 20 2013 *)

PARI

A096309=n->if(n>1,vecmax(apply(a,factor(n)[,2])))+1 \\ M. F. Hasler, Nov 21 2013

Extensions

More terms from Jean-François Alcover, Nov 20 2013