A130897 - OEIS

%I #57 Sep 10 2015 04:50:25

%S 16,48,80,81,112,144,162,176,208,240,256,272,304,324,336,368,400,405,

%T 432,464,496,512,528,560,567,592,624,625,648,656,688,720,752,768,784,

%U 810,816,848,880,891,912,944,976

%N Numbers that are not exponentially squarefree.

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

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

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

%C Product_{k = 1..A001221(n)} A008966(A124010(n,k)) = 0. - _Reinhard Zumkeller_, Mar 13 2012

%C The density of {a(n)} is 0.04407699... (see comment in A209061). - _Peter J. C. Moses_ and _Vladimir Shevelev_, Sep 08 2015

%H Reinhard Zumkeller, <a href="/A130897/b130897.txt">Table of n, a(n) for n = 1..10000</a>

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

%p filter:=n ->  not andmap(t -> numtheory:-issqrfree(t[2]), ifactors(n)[2]);

%p select(filter, [$1..1000]); # _Robert Israel_, Sep 03 2015

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

%o (Haskell)

%o a130897 n = a130897_list !! (n-1)

%o a130897_list = filter

%o    (any (== 0) . map (a008966 . fromIntegral) . a124010_row) [1..]

%o -- _Reinhard Zumkeller_, Mar 13 2012

%o (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 Complement of A209061; subsequence of A013929, A046099, and A046101.

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

%K nonn

%O 1,1

%A _Laszlo Toth_, Mar 18 2011