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A130897 Numbers that are not exponentially squarefree.

%I A130897 #57 Sep 10 2015 04:50:25
%S A130897 16,48,80,81,112,144,162,176,208,240,256,272,304,324,336,368,400,405,
%T A130897 432,464,496,512,528,560,567,592,624,625,648,656,688,720,752,768,784,
%U A130897 810,816,848,880,891,912,944,976
%N A130897 Numbers that are not exponentially squarefree.
%C A130897 A positive integer is called exponentially squarefree (e-squarefree) if in its prime power factorization all the exponents are squarefree.
%C A130897 a(n) is the sequence of positive integers in which prime power factorization there is at least one nonsquarefree exponent.
%C A130897 n is non-e-squarefree iff f(n)=0, where f(n) is the exponential Moebius function A166234.
%C A130897 Product_{k = 1..A001221(n)} A008966(A124010(n,k)) = 0. - _Reinhard Zumkeller_, Mar 13 2012
%C A130897 The density of {a(n)} is 0.04407699... (see comment in A209061). - _Peter J. C. Moses_ and _Vladimir Shevelev_, Sep 08 2015
%H A130897 Reinhard Zumkeller, <a href="/A130897/b130897.txt">Table of n, a(n) for n = 1..10000</a>
%H A130897 M. V. Subbarao, <a href="http://www.math.ualberta.ca/~subbarao/documents/Subbarao7.pdf">On some arithmetic convolutions</a>, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, 247-271, Springer, 1972, doi:10.1007/BFb0058796.
%H A130897 Laszlo Toth, <a href="http://arxiv.org/abs/0708.3557">On certain arithmetic functions involving exponential divisors, II.</a>, Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166.
%e A130897 16=2^4, 48=2^4*3, 256=2^8 are non-e-squarefree, since 4 and 8 are nonsquarefree.
%p A130897 filter:=n ->  not andmap(t -> numtheory:-issqrfree(t[2]), ifactors(n)[2]);
%p A130897 select(filter, [$1..1000]); # _Robert Israel_, Sep 03 2015
%t A130897 Select[Range@ 1000, ! AllTrue[Last /@ FactorInteger@ #, SquareFreeQ] &] (* _Michael De Vlieger_, Sep 07 2015, Version 10 *)
%o A130897 (Haskell)
%o A130897 a130897 n = a130897_list !! (n-1)
%o A130897 a130897_list = filter
%o A130897    (any (== 0) . map (a008966 . fromIntegral) . a124010_row) [1..]
%o A130897 -- _Reinhard Zumkeller_, Mar 13 2012
%o A130897 (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
%Y A130897 Complement of A209061; subsequence of A013929, A046099, and A046101.
%Y A130897 Cf. A005117, A166234, A049419, A051377, A008966, A124010.
%K A130897 nonn
%O A130897 1,1
%A A130897 _Laszlo Toth_, Mar 18 2011