A144972 Power-6-free numbers: numbers whose exponents in their prime factorization are all less than 6.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset

1,2

Comments

Superset of A005117 and A067259. The first numbers not in the sequence are 64, 128, 192, 256, 320, 384, 448, 512, 576, 640, 704, 729 etc. [R. J. Mathar, Oct 11 2008]

This sequence has an asymptotic density of about 0.98270. - David A. Corneth, Nov 05 2017

From Amiram Eldar, Mar 20 2021: (Start)

The asymptotic density of this sequence is 1/zeta(6) = 1/A013664 = 945/Pi^6 = 0.9829525922...

The Schnirelmann density of this sequence is 6165/6272 (Orr, 1969). (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.

Formula

{n: A051903(n) <= 5}. [R. J. Mathar, Oct 11 2008, corrected Oct 19 2008]

Maple

select(n -> max(seq(f[2], f=ifactors(n)[2]))<=5, [$1..1000]); # Robert Israel, Nov 05 2017

Mathematica

lst={}; Do[a=0; Do[If[FactorInteger[m][[n, 2]]>5, a=1], {n, Length[FactorInteger[m]]}]; If[a!=1, AppendTo[lst, m]], {m, 2*5!}]; lst
Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]] , #1 < 6 & ] &] (* Amiram Eldar, Mar 20 2021 *)

Prog

(Python)
from sympy import mobius, integer_nthroot
def A144972(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**6) for k in range(1, integer_nthroot(x, 6)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

Crossrefs

Subsequences: A005117, A004709, A046100, A067259, A051903.

Cf. A013664

Sequence in context: A130696 A146297 A296876 * A166719 A272159 A227981

Adjacent sequences: A144969 A144970 A144971 * A144973 A144974 A144975

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Sep 27 2008

Status

approved