A130897 Numbers that are not exponentially squarefree.

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

Offset: 1

Author

Laszlo Toth, Mar 18 2011

Keywords

• nonn

Comments

A positive integer is called exponentially squarefree (e-squarefree) if in its prime power factorization all the exponents are squarefree.

a(n) is the sequence of positive integers in which prime power factorization there is at least one nonsquarefree exponent.

n is non-e-squarefree iff f(n)=0, where f(n) is the exponential Moebius function A166234.

Product_{k = 1..A001221(n)} A008966(A124010(n,k)) = 0. - Reinhard Zumkeller, Mar 13 2012

The density of {a(n)} is 0.04407699... (see comment in A209061). - Peter J. C. Moses and Vladimir Shevelev, Sep 08 2015

Examples

			16=2^4, 48=2^4*3, 256=2^8 are non-e-squarefree, since 4 and 8 are nonsquarefree.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• M. V. Subbarao, On some arithmetic convolutions, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, 247-271, Springer, 1972, doi:10.1007/BFb0058796.
• Laszlo Toth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166.

Crossrefs

Complement of A209061; subsequence of A013929, A046099, and A046101.

Cf. A005117, A166234, A049419, A051377, A008966, A124010.

Programs

Haskell

a130897 n = a130897_list !! (n-1)
a130897_list = filter
   (any (== 0) . map (a008966 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Mar 13 2012

Maple

filter:=n ->  not andmap(t -> numtheory:-issqrfree(t[2]), ifactors(n)[2]);
select(filter, [$1..1000]); # Robert Israel, Sep 03 2015

Mathematica

Select[Range@ 1000, ! AllTrue[Last /@ FactorInteger@ #, SquareFreeQ] &] (* Michael De Vlieger, Sep 07 2015, Version 10 *)

PARI

is(n)=my(f=factor(n)[, 2]); for(i=1, #f, if(!issquarefree(f[i]), return(1))); 0 \\ Charles R Greathouse IV, Sep 03 2015