Search: id:A005117
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%I A005117 M0617
%S A005117 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,
               39,
%T A005117 41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,77,78,
               79,
%U A005117 82,83,85,86,87,89,91,93,94,95,97,101,102,103,105,106,107,109,110,111,
               113
%N A005117 Square-free numbers: numbers that are not divisible by a square greater 
               than 1.
%C A005117 1 together with the numbers that are products of distinct primes.
%C A005117 Also smallest sequence with the property that a(m)*a(n) is never a square 
               for n <> m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
%C A005117 Numbers n such that there is only one Abelian group with n elements, 
               the cyclic group of order n (the numbers such that A000688(n) = 1). 
               - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001
%C A005117 Numbers n such that A007913(n)>phi(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 10 2002
%C A005117 a(n) = smallest m with exactly n square-free numbers <= m. - Amarnath 
               Murthy (amarnath_murthy(AT)yahoo.com), May 21 2002
%C A005117 n is squarefree <=> n divides n# where n# = product of first n prime 
               numbers - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004
%C A005117 Numbers n such that omega(n)=Omega(n)=A072047(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jul 11 2006
%C A005117 The lcm of any subsequence of a(n) is in a(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jul 11 2006
%C A005117 Comment from Ed Pegg Jr (ed(AT)mathpuzzle.com), Jul 22 2008: This sequence 
               and the Beatty Pi^2/6 sequence (A059535) are "incestuous": the first 
               20000 terms are bounded within (-9, 14).
%C A005117 Let us introduce a function D(n)=sigma_0(n)/(2^(alfa(1)+...+alfa(r)), 
               sigma_0(n) number of divisors of n (A000005), prime factorization 
               of n=p(1)^alfa(1) * ... * p(r)^alfa(r), alfa(1)+...+alfa(r) is sequence 
               (A086436). Function D(n) splits the set of positive integers into 
               subsets, according to the value of D(n). Squarefree numbers (A005117) 
               has D(n)=1, other numbers are "deviated" from the squarefree ideal 
               and have 0<D(n)<1. For D(n)=1/2 we have A048109, for D(n)=3/4 we 
               have A067295. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Sep 21 
               2008]
%C A005117 Unit together with the numbers with equal number of prime and nonprime 
               divisors. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 
               10 2009]
%D A005117 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, 
               Paris 2008.
%D A005117 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 
               2nd ed., Wiley, NY, 1966, p. 251.
%D A005117 M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge 
               Univ. Press, page 432.
%D A005117 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005117 Daniel Forgues, <a href="b005117.txt">Table of n, a(n) for n=1..60794</
               a>
%H A005117 A. Granville, <a href="http://www.dms.umontreal.ca/~andrew/Postscript/
               polysq3.ps">ABC means we can count squarefrees</a>, International 
               Mathematical Research Notices 19 (1998), 991-1009.
%H A005117 A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               SquareFreeNumber.html">square-free number</a>
%H A005117 L. Marmet, <a href="http://www.marmet.org/louis/sqfgap/">First occurrences 
               of square-free gaps...</a>
%H A005117 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
               Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc. 5 
               (1913) 105-106.
%H A005117 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Squarefree.html">Link to a section of The World of Mathematics.</
               a>
%H A005117 Wikipedia, <a href="http://www.wikipedia.org/wiki/Square-free">Square-free</
               a>
%F A005117 Lim n -> infinity a(n)/n=Pi^2/6 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 23 2002
%F A005117 A039956 UNION A056911. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               May 16 2008
%p A005117 with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then 
               a := [ op(a), n ]; fi; od:
%t A005117 (* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) 
               Select[ Range[ 113], SquareFreeQ[ # ] &] (from Robert G. Wilson v 
               Jan 31 2005)
%t A005117 Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] - Joseph Biberstine 
               (jrbibers(AT)indiana.edu), Dec 26 2006
%t A005117 max = 0; a = {}; Do[m = FactorInteger[n]; w = Product[m[[k]][[1]], {k, 
               1, Length[m]}]; If[w > max, AppendTo[a, n]; max = w], {n, 1, 1000}]; 
               a - Artur Jasinski (grafix(AT)csl.pl), Apr 06 2008
%o A005117 (MAGMA) [ n : n in [1..1000] | IsSquarefree(n) ];
%o A005117 (PARI) bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),
               L[j]=i:j=j+1)); L
%o A005117 (PARI) {a(n)= local(m,c); if(n<=1,n==1, c=1; m=1; while( c<n, m++; if(issquarefree(m), 
               c++)); m)} /* Michael Somos Apr 29 2005 */
%Y A005117 Complement of A013929.
%Y A005117 Cf. A048640, A053797, A039956, A056911, A000924, A033197, A020753, A020754, 
               A020755, A000688, A003277.
%Y A005117 Cf. A013928.
%Y A005117 Cf. A136742, A136743, A072284, A120992.
%Y A005117 Cf. A160764 N_th square-free number minus round(n*zeta(2)). [From Daniel 
               Forgues (squid(AT)zensearch.com), May 26 2009]
%Y A005117 Sequence in context: A064052 A064594 A076144 this_sequence A144338 A077377 
               A076786
%Y A005117 Adjacent sequences: A005114 A005115 A005116 this_sequence A005118 A005119 
               A005120
%K A005117 nonn,easy,nice,core,new
%O A005117 1,2
%A A005117 N. J. A. Sloane (njas(AT)research.att.com).

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