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 A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n. 503

%I

%S 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,

%T 29,30,31,2,33,34,35,6,37,38,39,10,41,42,43,22,15,46,47,6,7,10,51,26,

%U 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

%N Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

%C Multiplicative with a(p^e) = p.

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

%C a(k)=k for k=squarefree numbers A005117. - _Lekraj Beedassy_, Sep 05 2006

%C 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.

%C a(n) = A128651(A129132(n-1) + 2) for n > 1. - _Reinhard Zumkeller_, Mar 30 2007

%C Also the least common multiple of the prime factors of n. - _Peter Luschny_, Mar 22 2011

%C The Mobius transform of the sequence generates the sequence of absolute values of A097945. - _R. J. Mathar_, Apr 04 2011

%C 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

%C a(n) differs from A014963(n) when n is a term of A024619. - _Eric Desbiaux_, Mar 24 2014

%C a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length. For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - _Lee A. Newberg_, Jul 27 2016

%C a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - _Juri-Stepan Gerasimov_, Apr 06 2017

%H T. D. Noe and Daniel Forgues, <a href="/A007947/b007947.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from T. D. Noe)

%H Masum Billal, <a href="http://arxiv.org/abs/1501.00609">Divisible Sequence and its Characteristic Sequence</a>, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.

%H Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>

%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]

%H Jarosław Grytczuk, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.039">Thue type problems for graphs, points and numbers</a>, Discrete Math., 308 (2008), 4419-4429.

%H Neville Holmes, <a href="http://www.comp.utas.edu.au/users/nholmes/sqncs/index.htm#A007947">Integer Sequences</a> [Broken link]

%H Serge Lang, <a href="http://dx.doi.org/10.1090/S0273-0979-1990-15899-9">Old and New Conjectured Diophantine Inequalities</a>, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.

%H D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf">Euler constants for arithmetical progressions</a>, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.

%H Ivar Peterson, <a href="http://web.archive.org/web/20130702061828/http://www.maa.org/mathland/mathtrek_12_8.html">The Amazing ABC Conjecture</a>

%H Paul Tarau, <a href="http://dx.doi.org/10.1007/978-3-642-23283-1_15">Emulating Primality with Multiset Representations of Natural Numbers</a>, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238

%H Paul Tarau, <a href="http://dx.doi.org/10.1016/j.tcs.2014.04.025">Towards a generic view of primality through multiset decompositions of natural numbers</a>, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.

%F If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).

%F a(n) = Product_{k=1..A001221(n)} A027748(n,k). - _Reinhard Zumkeller_, Aug 27 2011

%F Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - _R. J. Mathar_, Jan 21 2012

%F a(n) = Sum_{d|n} phi(d) * mu(d)^2. - _Enrique Pérez Herrero_, Apr 23 2012

%F a(n) = Product_{d|n} d^(moebius(n/d)) (see Billal link). - _Michel Marcus_, Jan 06 2015

%F a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - _Anthony Browne_, Jun 17 2016

%F a(n) = n/A003557(n). - _Juri-Stepan Gerasimov_, Apr 07 2017

%F G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Apr 11 2017

%F From _Antti Karttunen_, Jun 18 2017: (Start)

%F a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).

%F a(n) = A019565(A087207(n)). (End)

%e G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

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

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

%p seq(A007947(i),i=1..69); # _Peter Luschny_, Mar 22 2011

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

%p seq(A(n),n=1..100); # _Robert Israel_, Aug 10 2014

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

%t Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* _Harvey P. Dale_, Jul 14 2014 *)

%t a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* _Michael Somos_, Jul 15 2018 *)

%o (PARI) a(n) = factorback(factorint(n)[,1]); \\ _Andrew Lelechenko_, May 09 2014

%o (MAGMA) [ &*PrimeDivisors(n): n in [1..100] ]; // _Klaus Brockhaus_, Dec 04 2008

%o a007947 = product . a027748_row -- _Reinhard Zumkeller_, Feb 27 2012

%o (Sage) def A007947(n) : return mul(p for p in prime_divisors(n))

%o [A007947(n) for n in (1..60)] # _Peter Luschny_, Mar 07 2017

%o (Python)

%o from operator import mul

%o from sympy import primefactors

%o def a(n): return 1 if n<2 else reduce(mul, primefactors(n))

%o print [a(n) for n in xrange(1, 101)] # _Indranil Ghosh_, Apr 16 2017

%o (Scheme) (define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - _Antti Karttunen_, Jun 18 2017

%Y Cf. A048803, A007913, A020639, A028234, A062953, A000188, A019554, A019565, A020500, A053462, A010051, A284318, A000005, A001221, A005361, A003557, A034444, A060735, A065642, A066503, A079277, A087207.

%Y Bisections: A099984, A099985.

%Y Cf. also array A284311 (A284457).

%K nonn,easy,nice,mult

%O 1,2

%A R. Muller, Mar 15 1996

%E More terms from several people including _David W. Wilson_

%E Definition expanded by _Jonathan Sondow_, Apr 26 2013

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Last modified February 17 02:33 EST 2019. Contains 320200 sequences. (Running on oeis4.)