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A007913
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Squarefree part of n: a(n) = smallest positive number m such that n/m is a square.
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101
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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, 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, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also called core(n).
Sequence read mod 4 gives A065882. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 28 2004
This is an arithmetic function and is undefined if n <= 0.
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.
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
a(n) = the smallest natural numbers 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). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Apr 26 2010]
Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011
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REFERENCES
| 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.
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
John M. Campbell, An Integral Representation of Kekule' Numbers, and Double Integrals Related to Smarandache Sequences, Arxiv preprint arXiv:1105.3399, 2011.
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LINKS
| Daniel Forgues, Table of n, a(n) for n=1..100000
K. Atanassov, On Some of Smarandache's Problems
H. Bottomley, Some Smarandache-type multiplicative sequences
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
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
Dirichlet g.f. zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
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MAPLE
| 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
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MATHEMATICA
| 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]
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]
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PROG
| (MAGMA) [ Squarefree(n) : n in [1..256] ]; (N. J. A. Sloane, Dec 23 2006)
(PARI) a(n)=core(n)
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CROSSREFS
| Cf. A000188, A002734, A117811, A007947, A019554.
Sequence in context: A055231 A160400 A072400 * A083346 A065883 A071975
Adjacent sequences: A007910 A007911 A007912 * A007914 A007915 A007916
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KEYWORD
| nonn,easy,mult,nice
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AUTHOR
| R. Muller
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EXTENSIONS
| More terms from Michael Somos, Nov 24, 2001
Definition corrected by Daniel Forgues (squid(AT)zensearch.com), Mar 24 2009
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