A322030 Numbers whose prime factors all have the same order of primeness.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 23, 25, 26, 27, 28, 29, 31, 32, 37, 38, 41, 43, 46, 47, 49, 51, 52, 53, 56, 58, 59, 61, 64, 67, 71, 73, 74, 76, 79, 81, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 121, 122, 123

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

			182 is in the sequence because its prime factors 2, 7, 13 all have order of primeness 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• 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, A322027, A322028.

Programs

Maple

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= proc(n) option remember; local k; for k from 1+`if`(n=1,
      0, a(n-1)) while nops(map(p, factorset(k)))>1 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2018

Mathematica

ordpri[n_]:=If[!PrimeQ[n],0,Length[NestWhileList[PrimePi,PrimePi[n],PrimeQ]]];
Select[Range[200],SameQ@@ordpri/@FactorInteger[#][[All,1]]&]