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%I #151 Jul 03 2020 14:55:05
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,
%T 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,
%U 52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69
%N Exponentially squarefree numbers.
%C Numbers having only squarefree exponents in their canonical prime factorization;
%C A166234(a(n)) <> 0;
%C Product_{k=1..A001221(n)} A008966(A124010(n,k)) = 1.
%C According to the formula of Theorem 3 [Toth], the density of the exponentially squarefree numbers is 0.9559230158619... - _Peter J. C. Moses_ and _Vladimir Shevelev_, Sep 10 2015
%C From _Vladimir Shevelev_, Sep 24 2015: (Start)
%C A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.
%C Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.
%C Let {u(n)} be the characteristic function of S. Then, for the density h=h(S) of the exponentially S-numbers, we have the representations
%C h(S) = Product_{prime p} Sum_{j in S}(p-1)/p^(j+1) = Product_{p} (1 + Sum_{j>=1} (u(j) - u(j-1))/p^j}. In particular, if S = {0,1}, then the exponentially S-numbers are squarefree numbers; if S consists of 0 and {2^k}_(k>=0}, then the exponentially S-numbers form A138302 (see [Shevelev], 2007); if S consists of 0 and squarefree numbers, then u(n)=|mu(n)|, where mu(n) is the Möbius function (A008683), we obtain the density h of the exponentially squarefree numbers (cf. Toth's link, Theorem 3); the calculation of h with a very high degree of accuracy belongs to _Juan Arias-de-Reyna_ (A262276). Note that if S contains 1, then h(S) >= 1/zeta(2) = 6/Pi^2; otherwise h(S) = 0. Indeed, in the latter case, the density of the sequence of exponentially S-numbers does not exceed the density of A001694, which equals 0. (End)
%H Reinhard Zumkeller, <a href="/A209061/b209061.txt">Table of n, a(n) for n = 1..10000</a>
%H X. Cao, W. Zhai, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Cao/cao4.html">Some arithmetic functions involving exponential divisors</a>, JIS 13 (2010) 10.3.7
%H Y.-F. S. Petermann, <a href="http://ac.inf.elte.hu/Vol_032_2010/143.pdf">Arithmetical functions involving exponential divisors: note on two papers by L. Toth</a>, Ann. Univ. Sci. Budapest, Sect. Comp. 32 (2010) 143-149
%H Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa126-3-1">Compact integers and factorials</a>, Acta Arithmetica 126:3 (2007), pp. 195-236.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1510.05914">Exponentially S-numbers</a>, arXiv:1510.05914 [math.NT], 2015.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1511.03860">Set of all densities of exponentially S-numbers</a>, arXiv:1511.03860 [math.NT], 2015.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1602.04244">A fast computation of density of exponentially S-numbers</a>, arXiv:1602.04244 [math.NT], 2016.
%H Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa8395-5-2016">S-exponential numbers</a>, Acta Arithmetica, Vol. 175(2016), 385-395.
%H H. M. Stark, <a href="http://dx.doi.org/10.1090/S0002-9939-1966-0199161-1">On the asymptotic density of the k-free integers</a>, Proc. Amer. Soc. 17 (1966), 1211-1214.
%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 and arXiv:0708.3557 [math.NT], 2007-2009.
%F One can prove that the principal term of Toth's asymptotics for the density of this sequence (cf. Toth's link, Theorem 3) equals also Product_{prime p}(Sum_{j in S}(p-1)/p^{j+1})*x, where S is the set of 0 and squarefree numbers. The remainder term O(x^(0.2+t)), where t>0 is arbitrarily small, was obtained by L. Toth while assuming the Riemann Hypothesis. - _Vladimir Shevelev_, Sep 12 2015
%t Select[Range@ 69, Times @@ Boole@ Map[SquareFreeQ, Last /@ FactorInteger@ #] > 0 &] (* _Michael De Vlieger_, Sep 07 2015 *)
%o (Haskell)
%o a209061 n = a209061_list !! (n-1)
%o a209061_list = filter
%o (all (== 1) . map (a008966 . fromIntegral) . a124010_row) [1..]
%o (PARI) is(n)=my(f=factor(n)[,2]); for(i=1,#f,if(!issquarefree(f[i]), return(0))); 1 \\ _Charles R Greathouse IV_, Sep 02 2015
%Y Complement of A130897.
%Y A005117, A004709, and A046100 are subsequences.
%Y Cf. A001694, A008683, A036537, A115063, A138302, A197680, A262276, A262675, A268335, A270428.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Mar 13 2012
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