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

1,1

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

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.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A002808, A322449.

Sequence in context: A223440 A223402 A260985 * A354181 A264164 A045047

Adjacent sequences: A322445 A322446 A322447 * A322449 A322450 A322451

Keyword

nonn

Author

Alois P. Heinz, Dec 08 2018

Status

approved