A322028 Number of distinct orders of primeness among the prime factors of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 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 factors of 105 = 3*5*7 have 3 different orders of primeness, namely 2, 3, and 1 respectively.
		

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, A322027, A322030.

Programs

Maple

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

Mathematica

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