1,2

1 together with the numbers that are products of distinct primes.

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

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

Numbers n such that A007913(n)>phi(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 10 2002

a(n) = smallest m with exactly n square-free numbers <= m. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 21 2002

n is squarefree <=> n divides n# where n# = product of first n prime numbers - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004

Numbers n such that omega(n)=Omega(n)=A072047(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 11 2006

The lcm of any subsequence of a(n) is in a(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 11 2006

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).

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]

Unit together with the numbers with equal number of prime and nonprime divisors. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 10 2009]

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris 2008.

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.

M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n=1..60794

A. Granville, ABC means we can count squarefrees, International Mathematical Research Notices 19 (1998), 991-1009.

A. Krowne, PlanetMath.org, square-free number

L. Marmet, First occurrences of square-free gaps...

S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Square-free

Lim n -> infinity a(n)/n=Pi^2/6 - Benoit Cloitre (benoit7848c(AT)orange.fr), May 23 2002

A039956 UNION A056911. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 16 2008

with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:

(* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) Select[ Range[ 113], SquareFreeQ[ # ] &] (from Robert G. Wilson v Jan 31 2005)

Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006

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

(MAGMA) [ n : n in [1..1000] | IsSquarefree(n) ];

(PARI) bnd = 1000; L = vector(bnd); j = 1; for (i=1, bnd, if(issquarefree(i), L[j]=i:j=j+1)); L

(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 */

Complement of A013929.

Cf. A048640, A053797, A039956, A056911, A000924, A033197, A020753, A020754, A020755, A000688, A003277.

Cf. A013928.

Cf. A136742, A136743, A072284, A120992.

Cf. A160764 N_th square-free number minus round(n*zeta(2)). [From Daniel Forgues (squid(AT)zensearch.com), May 26 2009]

Sequence in context: A064052 A064594 A076144 this_sequence A144338 A077377 A076786

Adjacent sequences: A005114 A005115 A005116 this_sequence A005118 A005119 A005120

nonn,easy,nice,core,new

N. J. A. Sloane (njas(AT)research.att.com).