A046097 Values of n for which binomial(2n-1, n) is squarefree.

1, 2, 3, 4, 6, 9, 10, 12, 36

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• fini

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

• 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), 73-107.
• J. W. Sander, Prime power divisors of binomial coefficients, Journal für die reine und angewandte Mathematik 430 (1992), 1-20.

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 next-to-center column should be nonzero in A034931 (Pascal's triangle mod 4) and all similarly constructed fractal triangles (Pascal's triangle mod p^2).

Programs

Magma

[n: n in [1..150] | IsSquarefree(Binomial(2*n-1,n))]; // Vincenzo Librandi, Dec 10 2015

Maple

select(n -> numtheory:-issqrfree(binomial(2*n-1,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..j-1),j=1..N-1)}:
for i from 2 to 10 do
  Cands:= select(n -> carries(n-1,n,ithprime(i)) <= 1, Cands)
od:
select(n -> numtheory:-issqrfree(binomial(2*n-1,n)),Cands); # Robert Israel, Dec 10 2015

Mathematica

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

PARI

is(n)=issquarefree(binomial(2*n-1,n)) \\ Anders Hellström, Dec 09 2015

Extensions

James 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).