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A007947
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Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.
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292
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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J. Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
S. Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
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LINKS
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T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
H. Bottomley, Some Smarandache-type multiplicative sequences
S. R. Finch, Unitarism and infinitarism.
Neville Holmes, Integer Sequences
I. 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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
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PROG
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(PARI) a(n)=local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i])
(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, June 10 2012
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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R. Muller
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EXTENSIONS
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More terms from several people including David W. Wilson.
Definition expanded by Jonathan Sondow, Apr 26 2013
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STATUS
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approved
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