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A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square. 209
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). [Not to be confused with the squarefree kernel of n, A007947.]

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(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]

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

The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - Álvar Ibeas, Mar 19 2015

a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - Zak Seidov, Jan 30 2016

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

REFERENCES

K. Atanassov, On the 22nd, 23rd, and the 24th 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 Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 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

a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - Anthony Browne, Jun 06 2016

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

From Peter Munn, Nov 18 2019: (Start)

a(k*m) = A059897(a(k), a(m)).

a(n) = n / A008833(n).

(End)

a(A225546(n)) = A225546(A006519(n)). - Peter Munn, Jan 04 2020

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

# second Maple program:

a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):

seq(a(n), n=1..100);  # Alois P. Heinz, Jul 20 2015

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 *)

√#/.(c_:1)*a_^(b_:0)->(c*a^b)^2&/@Range@100 (* Bill Gosper, Jul 18 2015 *)

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

(Python)

from sympy import factorint, prod

def A007913(n):

    return prod(p for p, e in factorint(n).items() if e % 2)

# Chai Wah Wu, Feb 03 2015

(Sage)

[squarefree_part(n) for n in (1..77)] # Peter Luschny, Feb 04 2015

CROSSREFS

See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.

See A027746, A027748, A124010 for factorization data for n.

Analogous sequences: A050985, A053165, A055231.

Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums).

Related to A006519 via A225546.

Sequence in context: A325978 A326049 A072400 * A083346 A319652 A327938

Adjacent sequences:  A007910 A007911 A007912 * A007914 A007915 A007916

KEYWORD

nonn,easy,mult,nice

AUTHOR

R. Muller, Mar 15 1996

EXTENSIONS

More terms from Michael Somos, Nov 24 2001

Definition reformulated by Daniel Forgues, Mar 24 2009

STATUS

approved

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Last modified July 8 20:50 EDT 2020. Contains 335535 sequences. (Running on oeis4.)