%I #189 Aug 03 2024 07:10:33
%S 1,2,3,1,5,6,7,2,1,10,11,3,13,14,15,1,17,2,19,5,21,22,23,6,1,26,3,7,
%T 29,30,31,2,33,34,35,1,37,38,39,10,41,42,43,11,5,46,47,3,1,2,51,13,53,
%U 6,55,14,57,58,59,15,61,62,7,1,65,66,67,17,69,70,71,2,73,74,3,19,77
%N Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.
%C Also called core(n). [Not to be confused with the squarefree kernel of n, A007947.]
%C Sequence read mod 4 gives A065882. - _Philippe Deléham_, Mar 28 2004
%C This is an arithmetic function and is undefined if n <= 0.
%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(A007947(b),c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n. [Corrected by _M. F. Hasler_, Mar 01 2018]
%C If n > 1, the quantity f(n) = log(n/core(n))/log(n) satisfies 0 <= f(n) <= 1; f(n) = 0 when n is squarefree and f(n) = 1 when n is a perfect square. One can define n as being "epsilon-almost squarefree" if f(n) < epsilon. - Kurt Foster (drsardonicus(AT)earthlink.net), Jun 28 2008
%C a(n) is the smallest natural number m such that product of geometric mean of the divisors of n and geometric mean of the divisors of m are integers. Geometric mean of the divisors of number n is real number b(n) = Sqrt(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A000290: a(A000290(n)) = 1. For n = 8; b(8) = sqrt(8), a(n) = 2 because b(2) = sqrt(2); sqrt(8) * sqrt(2) = 4 (integer). - _Jaroslav Krizek_, Apr 26 2010
%C Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - _R. J. Mathar_, Feb 11 2011
%C Booker, Hiary, & Keating outline a method for bounding (on the GRH) a(n) for large n using L-functions. - _Charles R Greathouse IV_, Feb 01 2013
%C According to the formula a(n) = n/A000188(n)^2, the scatterplot exhibits the straight lines y=x, y=x/4, y=x/9, ..., i.e., y=x/k^2 for all k=1,2,3,... - _M. F. Hasler_, May 08 2014
%C The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - _Álvar Ibeas_, Mar 19 2015
%C a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - _Zak Seidov_, Jan 30 2016
%C All solutions of the Diophantine equation n*x=y^2 or, equivalently, G(n,x)=y, with G being the geometric mean, are of the form x=k^2*a(n), y=k*sqrt(n*a(n)), where k is a positive integer. - _Stanislav Sykora_, Feb 03 2016
%C If f is a multiplicative function then Sum_{d divides n} f(a(d)) is also multiplicative. For example, A010052(n) = Sum_{d divides n} mu(a(d)) and A046951(n) = Sum_{d divides n} mu(a(d)^2). - _Peter Bala_, Jan 24 2024
%H Daniel Forgues, <a href="/A007913/b007913.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 terms from T. D. Noe)
%H Krassimir T. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">On Some of Smarandache's Problems</a>.
%H Krassimir T. Atanassov, <a href="https://nntdm.net/volume-05-1999/number-2/80-82/">On the 22nd, 23rd, and the 24th Smarandache Problems</a>, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.
%H Andrew Booker, Ghaith Hiary, and Jon Keating, <a href="https://web.archive.org/web/20150912151134/http://www.cs.uleth.ca/~cnta2012/slides-cnta12/Andrew-Booker.pdf">Detecting squarefree numbers</a>, CNTA XII (2012).
%H Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.
%H John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
%H Vlad Copil and Laurenţiu Panaitopol, <a href="https://www.jstor.org/stable/43679320">Properties of a sequence generated by positive integers</a>, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 2 (2007), pp. 131-137; <a href="https://www.ssmr.ro/bulletin/volumes/50-2/node5.html">alternative link</a>.
%H Florentin Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, Xiquan Publ., Phoenix-Chicago, 1993.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.
%F Multiplicative with a(p^k) = p^(k mod 2). - _David W. Wilson_, Aug 01 2001
%F a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - _Philippe Deléham_, Mar 28 2004
%F Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - _R. J. Mathar_, Feb 11 2011
%F a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - _Anthony Browne_, Jun 06 2016
%F a(n) = rad(n)/a(n/rad(n)), where rad = A007947. This recurrence relation together with a(1) = 1 generate the sequence. - _Velin Yanev_, Sep 19 2017
%F From _Peter Munn_, Nov 18 2019: (Start)
%F a(k*m) = A059897(a(k), a(m)).
%F a(n) = n / A008833(n).
%F (End)
%F a(A225546(n)) = A225546(A006519(n)). - _Peter Munn_, Jan 04 2020
%F From _Amiram Eldar_, Mar 14 2021: (Start)
%F Theorems proven by Copil and Panaitopol (2007):
%F Lim sup_{n->oo} a(n+1)-a(n) = oo.
%F Lim inf_{n->oo} a(n+1)-a(n) = -oo.
%F Sum_{k=1..n} 1/a(k) ~ c*sqrt(n) + O(log(n)), where c = zeta(3/2)/zeta(3) (A090699). (End)
%F a(n) = A019554(n)^2/n. - _Jianing Song_, May 08 2022
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - _Amiram Eldar_, Oct 25 2022
%F a(n) = A007947(A350389(n)). - _Amiram Eldar_, Jan 20 2024
%p A007913 := proc(n) local f,a,d; f := ifactors(n)[2] ; a := 1 ; for d in f do if type(op(2,d),'odd') then a := a*op(1,d) ; end if; end do: a; end proc: # _R. J. Mathar_, Mar 18 2011
%p # second Maple program:
%p a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jul 20 2015
%p seq(n / expand(numtheory:-nthpow(n, 2)), n=1..77); # _Peter Luschny_, Jul 12 2022
%t data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[_] -> 1; sfp = data/sp /. Sqrt[x_] -> x (* _Artur Jasinski_, Nov 03 2008 *)
%t Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,100}] (* _Zak Seidov_, Apr 08 2009 *)
%t Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* _T. D. Noe_, May 20 2013 *)
%t Sqrt[#] /. (c_:1)*a_^(b_:0) -> (c*a^b)^2& /@ Range@100 (* _Bill Gosper_, Jul 18 2015 *)
%o (Magma) [ Squarefree(n) : n in [1..256] ]; // _N. J. A. Sloane_, Dec 23 2006
%o (PARI) a(n)=core(n)
%o (Haskell)
%o a007913 n = product $
%o zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)
%o -- _Reinhard Zumkeller_, Jul 06 2012
%o (Python)
%o from sympy import factorint, prod
%o def A007913(n):
%o return prod(p for p, e in factorint(n).items() if e % 2)
%o # _Chai Wah Wu_, Feb 03 2015
%o (Sage)
%o [squarefree_part(n) for n in (1..77)] # _Peter Luschny_, Feb 04 2015
%Y See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.
%Y See A027746, A027748, A124010 for factorization data for n.
%Y Analogous sequences: A050985, A053165, A055231.
%Y Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums), A090699, A350389.
%Y Related to A006519 via A225546.
%K nonn,easy,mult,nice
%O 1,2
%A R. Muller, Mar 15 1996
%E More terms from _Michael Somos_, Nov 24 2001
%E Definition reformulated by _Daniel Forgues_, Mar 24 2009