OFFSET
0,4
COMMENTS
Binomial(2n,n) is squarefree for only n = 0, 1, 2, 4. Sequence A059097 lists n such that a(n) = 0 or 2. The plot shows the quadratic nature of this sequence. Sequence A110494 makes the quadratic behavior clearer.
Granville and Ramaré show that if n >= 2082 then a(n) >= sqrt(n/5). - Robert Israel, Sep 04 2019
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
T. D. Noe, Plot of A110493
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI].
EXAMPLE
a(5) = 3 because binomial(10,5) = 252 = (2^2)(3^2)(7).
MAPLE
f:= proc(n) local F;
F:= select(t -> t[2]>=2, ifactors(binomial(2*n, n))[2]);
if F = [] then 0 else max(map(t -> t[1], F)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Sep 04 2019
MATHEMATICA
Table[f=FactorInteger[Binomial[2n, n]]; s=Select[f, #[[2]]>1&]; If[s=={}, 0, s[[-1, 1]]], {n, 0, 100}]
CROSSREFS
KEYWORD
nonn,look
AUTHOR
T. D. Noe, Jul 22 2005
EXTENSIONS
a(0) prepended by T. D. Noe, Mar 27 2014
STATUS
approved