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Search: id:A005117
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| A005117 |
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Square-free numbers.
(Formerly M0617)
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+0
266
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| 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, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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
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
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REFERENCES
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
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
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FORMULA
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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
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MAPLE
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with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
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MATHEMATICA
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(* 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*) - Artur Jasinski (grafix(AT)csl.pl), Apr 06 2008
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PROGRAM
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(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 */
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CROSSREFS
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Cf. A048640, A053797, A039956, A056911, A000924, A033197, A020753, A020754, A020755, A000688, A003277.
Cf. A013928.
Complement of A013929.
Cf. A136742, A136743, A072284, A120992.
Sequence in context: A064052 A064594 A076144 this_sequence A077377 A076786 A087008
Adjacent sequences: A005114 A005115 A005116 this_sequence A005118 A005119 A005120
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KEYWORD
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nonn,easy,nice,core
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AUTHOR
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njas
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