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A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n. 308
1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Multiplicative with a(p^e) = p.

For n>1, product of the distinct prime factors of n.

a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006

A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), LCM(b,c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n.

a(n) = A128651(A129132(n-1) + 2) for n>1. - Reinhard Zumkeller, Mar 30 2007

Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011

The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011

Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013

a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014

LINKS

T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)

Henry Bottomley, Some Smarandache-type multiplicative sequences

Steven R. Finch, Unitarism and infinitarism.

Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.

Neville Holmes, Integer Sequences

Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.

Ivar Peterson, The Amazing ABC Conjecture

Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238

P. Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.

FORMULA

n = Product (p_j^k_j) -> Product (p_j).

a(n) = Product(A027748(n,k): 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011

Dirichlet g.f.: zeta(s)*product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012

a(n) = sum(d|n, phi(d) * mu(d)^2). - Enrique Pérez Herrero, Apr 23 2012

MAPLE

with(numtheory); A007947 := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1], i=1..nops(t1)); end;

A007947 := n -> ilcm(op(numtheory[factorset](n))):

seq(A007947(i), i=1..69); # Peter Luschny, Mar 22 2011

A:= n -> convert(numtheory:-factorset(n), `*`):

seq(A(n), n=1..100); # Robert Israel, Aug 10 2014

MATHEMATICA

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)

Table[Last[Select[Divisors[n], SquareFreeQ]], {n, 100}] (* Harvey P. Dale, Jul 14 2014 *)

PROG

(PARI) a(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014

(MAGMA) [ &*PrimeDivisors(n): n in [1..100] ]; /* Klaus Brockhaus, Dec 04 2008 */

(Haskell)

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

(Sage) def A007947(n) : return 1/mul(1/p for p in prime_divisors(n))

[A007947(n) for n in (1..60)] # Peter Luschny, Jun 10 2012

CROSSREFS

Cf. A048803, A007913, A062953, A000188, A019554, A020500, A053462.

Bisection: A099984, A099985.

Sequence in context: A056554 A088835 * A015053 A062953 A015052 A053166

Adjacent sequences:  A007944 A007945 A007946 * A007948 A007949 A007950

KEYWORD

nonn,easy,nice,mult

AUTHOR

R. Muller

EXTENSIONS

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

STATUS

approved

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Last modified October 25 12:11 EDT 2014. Contains 248527 sequences.