

A046097


Values of n for which binomial(2n1, n) is squarefree.


5




OFFSET

1,2


COMMENTS

No more terms up to 2^300. The sequence is finite by results of Sander and of Granville and Ramaré (see links).  Robert Israel, Dec 10 2015


LINKS

Table of n, a(n) for n=1..9.
Eric Weisstein's World of Mathematics, Binomial Coefficient.
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73107.
J. W. Sander, Prime power divisors of binomial coefficients, Journal für die reine und angewandte Mathematik 430 (1992), 120.


MAPLE

select(n > numtheory:issqrfree(binomial(2*n1, n)), [$1..2000]); # Robert Israel, Dec 09 2015
N:= 300: # to find all terms <= 2^N
carries:= proc(n, m, p)
# number of carries when adding n + m in base p.
local A, B, C, j, nc, t;
A:= convert(m, base, p);
B:= convert(n, base, p);
C:= 0; nc:= 0;
if nops(A) < nops(B) then A = [op(A), 0$(nops(B)nops(A))]
elif nops(A) > nops(B) then B:= [op(B), 0$(nops(A)nops(B))]
fi;
for j from 1 to nops(A) do
t:= C + A[j] + B[j];
if t >= p then
nc:= nc+1;
C:= 1;
else
C:= 0
fi
od:
nc;
end proc:
Cands:= {seq(2^j, j=0..N), seq(seq(2^j + 2^k, k=0..j1), j=1..N1)}:
for i from 2 to 10 do
Cands:= select(n > carries(n1, n, ithprime(i)) <= 1, Cands)
od:
select(n > numtheory:issqrfree(binomial(2*n1, n)), Cands); # Robert Israel, Dec 10 2015


MATHEMATICA

Select[ Range[1500], SquareFreeQ[ Binomial[ 2#1, #]] &] (* JeanFrançois Alcover, Oct 25 2012 *)


PROG

(PARI) is(n)=issquarefree(binomial(2*n1, n)) \\ Anders Hellström, Dec 09 2015
(MAGMA) [n: n in [1..150]  IsSquarefree(Binomial(2*n1, n))]; // Vincenzo Librandi, Dec 10 2015


CROSSREFS

Cf. A001700.
For a term to be here, it needs to be at least in the intersection of A048645, A051382, A050607, A050608 and an infinitude of similar sequences. The corresponding location in nexttocenter column should be nonzero in A034931 (Pascal's triangle mod 4) and all similarly constructed fractal triangles (Pascal's triangle mod p^2).
Sequence in context: A177919 A128399 A051404 * A239580 A175515 A241241
Adjacent sequences: A046094 A046095 A046096 * A046098 A046099 A046100


KEYWORD

nonn,fini


AUTHOR

Eric W. Weisstein


EXTENSIONS

James A. Sellers reports no further terms below 1500.
Michael Somos checked to 99999. Probably there are no more terms.
Mauro Fiorentini checked up to 2^64, as for n = 545259520, the binomial coefficient is a multiple of 5^4 and other possible exceptions have been checked (see Weisstein page for details).


STATUS

approved



