A322448 Numbers whose prime factorization contains at least one composite exponent.

16, 48, 64, 80, 81, 112, 144, 162, 176, 192, 208, 240, 256, 272, 304, 320, 324, 336, 368, 400, 405, 432, 448, 464, 496, 512, 528, 560, 567, 576, 592, 624, 625, 648, 656, 688, 704, 720, 729, 752, 768, 784, 810, 816, 832, 848, 880, 891, 912, 944, 960, 976, 1008

Offset: 1

Author

Alois P. Heinz, Dec 08 2018

Keywords

• nonn

Comments

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + Sum_{q prime >= 5} 1/p^q - 1/p^(q-1)) = 0.05328066264472198953... (using the method of Shevelev, 2016). - Amiram Eldar, Nov 08 2020

Examples

			16 = 2^4 is a term because 4 is a composite exponent here.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

Crossrefs

Cf. A002808, A322449.

Programs

Maple

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while andmap(i-> i[2]=1 or
       isprime(i[2]), ifactors(k)[2]) do od; k
    end:
seq(a(n), n=1..70);

Mathematica

Select[Range[1000], AnyTrue[FactorInteger[#][[;; , 2]], CompositeQ] &] (* Amiram Eldar, Nov 08 2020 *)

PARI

isok(m) = #select(x->((x>1) && !isprime(x)), factor(m)[,2]) > 0; \\ Michel Marcus, Dec 02 2020