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A007913 Squarefree part of n: a(n) = smallest positive number m such that n/m is a square. 110
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; text; internal format)
OFFSET

1,2

COMMENTS

Also called core(n).

Sequence read mod 4 gives A065882. - Philippe Deléham, 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). - Jaroslav Krizek, Apr 26 2010

Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011

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

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

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.

LINKS

T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)

K. Atanassov, On Some of Smarandache's Problems

Andrew Booker, Ghaith Hiary, and Jon Keating, Detecting squarefree numbers, CNTA XII (2012).

H. Bottomley, Some Smarandache-type multiplicative sequences

John M. Campbell, An Integral Representation of Kekule' Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399, 2011.

F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.

Eric Weisstein's World of Mathematics, Squarefree Part

FORMULA

Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson, Aug 01, 2001.

a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - Philippe Deléham, Mar 28 2004

Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011

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

MATHEMATICA

data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[_] -> 1; sfp = data/sp /. Sqrt[x_] -> x (* Artur Jasinski, Nov 03 2008 *)

Table[Times@@Power@@@({#[[1]], Mod[ #[[2]], 2]}&/@FactorInteger[n]), {n, 100}] (* Zak Seidov, Apr 08 2009 *)

Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* T. D. Noe, May 20 2013 *)

PROG

(MAGMA) [ Squarefree(n) : n in [1..256] ]; // N. J. A. Sloane, Dec 23 2006

(PARI) a(n)=core(n)

(Haskell)

a007913 n = product $

            zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)

-- Reinhard Zumkeller, Jul 06 2012

CROSSREFS

Cf. A000188, A002734, A117811, A007947, A019554, A027748, A124010.

Sequence in context: A055231 A160400 A072400 * A083346 A065883 A214392

Adjacent sequences:  A007910 A007911 A007912 * A007914 A007915 A007916

KEYWORD

nonn,easy,mult,nice,changed

AUTHOR

R. Muller, Mar 15 1996

EXTENSIONS

More terms from Michael Somos, Nov 24 2001

Definition changed by Daniel Forgues, Mar 24 2009

STATUS

approved

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Last modified October 23 03:24 EDT 2014. Contains 248411 sequences.