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A005117
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Squarefree numbers: numbers that are not divisible by a square greater than 1.
(Formerly M0617)
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401
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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
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OFFSET
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1,2
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COMMENTS
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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
Numbers n such that A007913(n)>phi(n) - Benoit Cloitre, Apr 10 2002
a(n) = smallest m with exactly n squarefree 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, Jul 11 2006
The lcm of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006
This sequence and the Beatty Pi^2/6 sequence (A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). [Ed Pegg Jr, Jul 22 2008]
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, Sep 21 2008]
A122840(a(n)) <= 1; A010888(a(n)) < 9. [From Reinhard Zumkeller, Mar 30 2010]
a(n)=A055229(A062838(n)) and a(n)>A055229(m) for m < A062838(n). [From Reinhard Zumkeller, Apr 09 2010]
Numbers n such that gcd(n,n')=1 where n' is the arithmetic derivative (A003415) of n. - Giorgio Balzarotti, Apr 23 2011
Numbers n such that A007913(n)=core(n)=n. [Franz Vrabec, Aug 27 2011]
Numbers n such that sqrt(n) cannot be simplified. [Sean Loughran, Sep 04 2011]
Indices where A057918(n)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is aquare. [John W. Layman, Sep 08 2011]
It appears that a(n) = n such that product(ithprime(k), k=1..n) mod n = 0. See Maple code. [From Gary Detlefs, Dec 07 2011]
A008477(a(n)) = 1. [Reinhard Zumkeller, Feb 17 2012]
A055653(a(n)) = a(n); A055654(a(n)) = 0. [Reinhard Zumkeller, Mar 11 2012]
A008966(a(n)) = 1. - Reinhard Zumkeller, May 26 2012
Sum(n>=1, 1/a(n)^s) = Zeta(s)/Zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012
A056170(a(n)) = 0. - Reinhard Zumkeller, Dec 29 2012
Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that sum_{k=1}^n a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). [Zhi-Wei Sun, Mar 26, 2013]
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris 2008.
P. Haukkanen, M. Mattila, J. K. Merikoski and T. Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, 16 (2013), #13.1.2. - From N. J. A. Sloane, Feb 03 2013
L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829, 2012. - From N. J. A. Sloane, Jan 01 2013
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).
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LINKS
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T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..60794 (first 10000 terms from T. D. Noe)
A. Granville, ABC means we can count squarefrees, International Mathematical Research Notices 19 (1998), 991-1009.
A. Krowne, PlanetMath.org, squarefree number
L. Marmet, First occurrences of squarefree gaps and an an algorithm for their computation
S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.
Eric Weisstein's World of Mathematics, Squarefree
Wikipedia, Squarefree integer
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FORMULA
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Lim n -> infinity a(n)/n=Pi^2/6 - Benoit Cloitre, May 23 2002
A039956 UNION A056911. - R. J. Mathar, 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:
t:= n-> product(ithprime(k), k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; [Gary Detlefs, Dec 07 2011]
A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Jan 09 2013
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MATHEMATICA
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Needs["NumberTheory`NumberTheoryFunctions`"]
Select[ Range[ 113], SquareFreeQ[ # ] &] (* 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, Apr 06 2008
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PROG
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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 */
(PARI) list(n)=my(v=vectorsmall(n, i, 1), u, j); forprime(p=2, sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1, n, v[i])); for(i=1, n, if(v[i], u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012
(Haskell)
a005117 n = a005117_list !! (n-1)
a005117_list = filter ((== 1) . a008966) [1..]
-- Reinhard Zumkeller, Aug 15 2011, May 10 2011
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CROSSREFS
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Complement of A013929. Subsequence of A209061.
Cf. A048640, A053797, A039956, A056911, A033197, A020753, A020754, A020755, A000688, A003277, A013928, A136742, A136743, A072284, A120992, A057918, A133466, A030059, A030229, A160764, A076259.
Sequence in context: A193304 A076144 * A144338 A077377 A076786 A167171
Adjacent sequences: A005114 A005115 A005116 * 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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N. J. A. Sloane.
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STATUS
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approved
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