A144972 - OEIS

%I #25 Aug 05 2024 16:04:17

%S 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,

%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,

%U 50,51,52,53,54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73

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

%C 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]

%C This sequence has an asymptotic density of about 0.98270. - _David A. Corneth_, Nov 05 2017

%C From _Amiram Eldar_, Mar 20 2021: (Start)

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

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

%H Robert Israel, <a href="/A144972/b144972.txt">Table of n, a(n) for n = 1..10000</a>

%H Richard C. Orr, <a href="https://doi.org/10.1112/jlms/s1-44.1.313">On the Schnirelmann density of the sequence of k-free integers</a>, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.

%F {n: A051903(n) <= 5}. [_R. J. Mathar_, Oct 11 2008, corrected Oct 19 2008]

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

%t 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

%t Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]] , #1 < 6 & ] &] (* _Amiram Eldar_, Mar 20 2021 *)

%o (Python)

%o from sympy import mobius, integer_nthroot

%o def A144972(n):

%o     def f(x): return n+x-sum(mobius(k)*(x//k**6) for k in range(1, integer_nthroot(x,6)[0]+1))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Aug 05 2024

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

%Y Cf. A013664

%K nonn

%O 1,2

%A _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008