A322027 Maximum order of primeness among the prime factors of n; a(1) = 0.

0, 1, 2, 1, 3, 2, 1, 1, 2, 3, 4, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 4, 1, 2, 3, 1, 2, 1, 1, 3, 5, 1, 4, 2, 3, 2, 1, 1, 2, 3, 2, 2, 1, 4, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 4, 1, 2, 1, 3, 3, 1, 5, 2, 1, 3, 4, 2, 2, 2, 3, 1, 2, 1, 1, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 3, 1, 2

Offset: 1

Author

Gus Wiseman, Nov 24 2018

Keywords

• nonn

Comments

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

Examples

			a(105) = 3 because the prime factor of 105 = 3*5*7 with maximum order of primeness is 5, with order 3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65536
• N. Fernandez, An order of primeness, F(p)
• N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Crossrefs

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322028, A322030.

Programs

Maple

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> max(0, map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Mathematica

Table[If[n==1,0,Max@@(Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]])],{n,100}]