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%I #39 Sep 08 2022 08:44:56
%S 1,2,3,4,6,9,10,12,36
%N Values of n for which binomial(2n-1, n) is squarefree.
%C 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
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialCoefficient.html">Binomial Coefficient.</a>
%H A. Granville and O. Ramaré, <a href="http://dx.doi.org/10.1112/S0025579300011608">Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients</a>, Mathematika 43 (1996), 73-107.
%H J. W. Sander, <a href="https://eudml.org/doc/153440">Prime power divisors of binomial coefficients</a>, Journal für die reine und angewandte Mathematik 430 (1992), 1-20.
%p select(n -> numtheory:-issqrfree(binomial(2*n-1,n)), [$1..2000]); # _Robert Israel_, Dec 09 2015
%p N:= 300: # to find all terms <= 2^N
%p carries:= proc(n,m,p)
%p # number of carries when adding n + m in base p.
%p local A,B,C,j,nc, t;
%p A:= convert(m,base,p);
%p B:= convert(n,base,p);
%p C:= 0; nc:= 0;
%p if nops(A) < nops(B) then A = [op(A),0$(nops(B)-nops(A))]
%p elif nops(A) > nops(B) then B:= [op(B), 0$(nops(A)-nops(B))]
%p fi;
%p for j from 1 to nops(A) do
%p t:= C + A[j] + B[j];
%p if t >= p then
%p nc:= nc+1;
%p C:= 1;
%p else
%p C:= 0
%p fi
%p od:
%p nc;
%p end proc:
%p Cands:= {seq(2^j,j=0..N), seq(seq(2^j + 2^k, k=0..j-1),j=1..N-1)}:
%p for i from 2 to 10 do
%p Cands:= select(n -> carries(n-1,n,ithprime(i)) <= 1, Cands)
%p od:
%p select(n -> numtheory:-issqrfree(binomial(2*n-1,n)),Cands); # _Robert Israel_, Dec 10 2015
%t Select[ Range[1500], SquareFreeQ[ Binomial[ 2#-1, #]] &] (* _Jean-François Alcover_, Oct 25 2012 *)
%o (PARI) is(n)=issquarefree(binomial(2*n-1,n)) \\ _Anders Hellström_, Dec 09 2015
%o (Magma) [n: n in [1..150] | IsSquarefree(Binomial(2*n-1,n))]; // _Vincenzo Librandi_, Dec 10 2015
%Y Cf. A001700.
%Y 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).
%K nonn,fini
%O 1,2
%A _Eric W. Weisstein_
%E _James A. Sellers_ reports no further terms below 1500.
%E _Michael Somos_ checked to 99999. Probably there are no more terms.
%E _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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