Search: id:A007913
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%I A007913
%S A007913 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 A007913 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 A007913 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 A007913 Square-free part of n: a(n) = smallest positive number m such that n/
               m is a square.
%C A007913 Also called core(n).
%C A007913 Sequence read mod 4 gives A065882. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 28 2004
%C A007913 This is an arithmetic function and is undefined if n <= 0.
%C A007913 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 A007913 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
%D A007913 K. Atanassov, On the 22-nd, the 23-th and the 24-th Smarandache Problems, 
               Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, 
               Vol. 5 (1999), No. 2, 80-82.
%D A007913 K. Atanassov, On Some of Smarandache's Problems, American Research Press, 
               1999, 16-21.
%D A007913 F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 
               1993.
%H A007913 Daniel Forgues, <a href="b007913.txt">Table of n, a(n) for n=1..100000</
               a>
%H A007913 K. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">
               On Some of Smarandache's Problems</a>
%H A007913 H. Bottomley, <a href="http://www.gallup.unm.edu/~smarandache/math.htm">
               Some Smarandache-type multiplicative sequences</a>
%H A007913 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">
               Only Problems, Not Solutions!</a>.
%H A007913 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquarefreePart.html">Link to a section of The World of Mathematics.</
               a>
%F A007913 Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson (davidwwilson(AT)comcast.net), 
               Aug 01, 2001.
%F A007913 a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. 
               - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 28 2004
%t A007913 data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[_] -> 1; sfp = 
               data/sp /. Sqrt[x_] -> x [From Artur Jasinski (grafix(AT)csl.pl), 
               Nov 03 2008]
%t A007913 Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,
               100}] (See Weisstein, Eric W."Square Part," http://mathworld.wolfram.com/
               SquarePart.html) [From Zak Seidov (zakseidov(AT)yahoo.com), Apr 08 
               2009]
%o A007913 (MAGMA) [ Squarefree(n) : n in [1..256] ]; (N. J. A. Sloane, Dec 23 2006)
%o A007913 (PARI) a(n)=core(n)
%Y A007913 Cf. A000188, A002734, A117811, A007947, A019554.
%Y A007913 Sequence in context: A055231 A160400 A072400 this_sequence A083346 A065883 
               A071975
%Y A007913 Adjacent sequences: A007910 A007911 A007912 this_sequence A007914 A007915 
               A007916
%K A007913 nonn,easy,mult,nice
%O A007913 1,2
%A A007913 R. Muller
%E A007913 More terms from Michael Somos, Nov 24, 2001
%E A007913 Definition corrected by Daniel Forgues (squid(AT)zensearch.com), Mar 
               24 2009

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