

A007913


Squarefree part of n: a(n) = smallest positive number m such that n/m is a square.


165



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
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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 "epsilonalmost 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 Lfunctions.  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, 8082.


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 Smarandachetype 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., PhoenixChicago, 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(s1)/zeta(2s2).  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


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 operator import mul
from functools import reduce
from sympy import factorint
def A007913(n):
....return reduce(mul, [1]+[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

Cf. A000188, A002734, A117811, A007947, A019554, A027748, A124010.
Sequence in context: A304339 A160400 A072400 * A083346 A319652 A065883
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



