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

%I A322028 #10 Nov 28 2018 18:58:10
%S A322028 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,
%T A322028 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,
%U A322028 2,2,1,2,1,1,2,1,2,2,1,2,1,2,1,2,2,1,2
%N A322028 Number of distinct orders of primeness among the prime factors of n.
%C A322028 The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.
%H A322028 Alois P. Heinz, <a href="/A322028/b322028.txt">Table of n, a(n) for n = 1..65536</a>
%H A322028 N. Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>
%H A322028 N. Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included at A006450 with permission of the author]
%e A322028 a(105) = 3 because the prime factors of 105 = 3*5*7 have 3 different orders of primeness, namely 2, 3, and 1 respectively.
%p A322028 with(numtheory):
%p A322028 p:= proc(n) option remember;
%p A322028       `if`(isprime(n), 1+p(pi(n)), 0)
%p A322028     end:
%p A322028 a:= n-> nops(map(p, factorset(n))):
%p A322028 seq(a(n), n=1..120);  # _Alois P. Heinz_, Nov 24 2018
%t A322028 Table[If[n==1,0,Length[Union[Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]]]]],{n,100}]
%Y A322028 Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322027, A322030.
%K A322028 nonn
%O A322028 1,6
%A A322028 _Gus Wiseman_, Nov 24 2018