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A046097 Values of n for which binomial(2n-1, n) is squarefree. 5
1, 2, 3, 4, 6, 9, 10, 12, 36 (list; graph; refs; listen; history; text; internal format)



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


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), 73-107.

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


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))]


for j from 1 to nops(A) do

    t:= C + A[j] + B[j];

    if t >= p then

       nc:= nc+1;

       C:= 1;


       C:= 0




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)


select(n -> numtheory:-issqrfree(binomial(2*n-1, n)), Cands); # Robert Israel, Dec 10 2015


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

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


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).

Sequence in context: A177919 A128399 A051404 * A239580 A337724 A175515

Adjacent sequences:  A046094 A046095 A046096 * A046098 A046099 A046100




Eric W. Weisstein


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).



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Last modified June 22 11:17 EDT 2021. Contains 345375 sequences. (Running on oeis4.)