A209061 Exponentially squarefree numbers.

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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset

1,2

Comments

Numbers having only squarefree exponents in their canonical prime factorization;

A166234(a(n)) <> 0;

Product_{k=1..A001221(n)} A008966(A124010(n,k)) = 1.

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

From Vladimir Shevelev, Sep 24 2015: (Start)

A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.

Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.

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

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)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

X. Cao, W. Zhai, Some arithmetic functions involving exponential divisors, JIS 13 (2010) 10.3.7

Y.-F. S. Petermann, Arithmetical functions involving exponential divisors: note on two papers by L. Toth, Ann. Univ. Sci. Budapest, Sect. Comp. 32 (2010) 143-149

Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126:3 (2007), pp. 195-236.

Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.

Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.

Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.

H. M. Stark, On the asymptotic density of the k-free integers, Proc. Amer. Soc. 17 (1966), 1211-1214.

Laszlo Toth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166 and arXiv:0708.3557 [math.NT], 2007-2009.

Formula

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

Mathematica

Select[Range@ 69, Times @@ Boole@ Map[SquareFreeQ, Last /@ FactorInteger@ #] > 0 &] (* Michael De Vlieger, Sep 07 2015 *)

Prog

(Haskell)
a209061 n = a209061_list !! (n-1)
a209061_list = filter
   (all (== 1) . map (a008966 . fromIntegral) . a124010_row) [1..]
(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

Crossrefs

Complement of A130897.

A005117, A004709, and A046100 are subsequences.

Cf. A001694, A008683, A036537, A115063, A138302, A197680, A262276, A262675, A268335, A270428.

Sequence in context: A288139 A194897 A140823 * A115063 A369939 A178210

Adjacent sequences: A209058 A209059 A209060 * A209062 A209063 A209064

Keyword

nonn

Author

Reinhard Zumkeller, Mar 13 2012

Status

approved