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A046097
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Values of n for which binomial(2n-1, n) is squarefree.
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5
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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MAPLE
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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
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MATHEMATICA
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PROG
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(Magma) [n: n in [1..150] | IsSquarefree(Binomial(2*n-1, n))]; // Vincenzo Librandi, Dec 10 2015
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CROSSREFS
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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).
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KEYWORD
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nonn,fini
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AUTHOR
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EXTENSIONS
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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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STATUS
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approved
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