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%I #74 Apr 26 2016 11:25:12
%S 1,2,3,5,7,11,23
%N Positive numbers n such that the numbers binomial(n,k) are squarefree for all k = 0..n.
%C It has been shown by Granville and Ramaré that the sequence is complete.
%C These are all the positive integers m that, when m is represented in binary, contain no composites represented in binary as substrings. - _Leroy Quet_, Oct 30 2008
%C This is a number-theoretic sequence, so it automatically assumes that n is positive. To quote Granville and Ramaré, "From Theorem 2 it is evident that there are only finitely many rows of Pascal's Triangle in which all of the entries are squarefree. In section 2 we show that this only occurs in rows 1, 2, 3, 5, 7, 11 and 23 (a result proved by Erdős long ago)." - _N. J. A. Sloane_, Mar 06 2014
%C See also comment in A249441. - _Vladimir Shevelev_, Oct 29 2014
%C This sequence is equivalent to: Positive integers n such that Fibonacci(n+1) divides n!. This comment depends on the finiteness of A019532. - _Altug Alkan_, Mar 31 2016
%H A. Granville and O. Ramaré, <a href="http://www.dms.umontreal.ca/~andrew/PDF/ramare.pdf">Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients</a>, Mathematika 43 (1996), 73-107, <a href="http://dx.doi.org/10.1112/S0025579300011608">[DOI]</a>.
%H H. N. Ramaswamy and R. Siddaramu, <a href="http://eprints.uni-mysore.ac.in/id/eprint/10232">On the Stufe, unit Stufe and Pythagoras number of the ring of integers modulo n</a>, Adv. Stud. Contemp. Math. (Kyungshang) 20:3 (2010), pp. 373-388.
%F Integers n>0 in set difference between union (A000225, A055010) and A249452. - _Vladimir Shevelev_, Oct 30 2014
%F a(n) = A018253(n+1) - 1. - _Altug Alkan_, Apr 26 2016
%e n=11: C[11,k] = 1, 11, 55, 165, 330, 462, ... are all squarefree (or 1).
%p select(n -> andmap(t -> numtheory:-issqrfree(binomial(n,t)),[$1..floor(n/2)]),[$1..100]); # _Robert Israel_, Oct 29 2014
%t Do[m = Prime[n]; k = 2; While[k < m/2 + .5 && Union[ Transpose[ FactorInteger[ Binomial[m, k]]] [[2]]] [[ -1]] < 2, k++ ]; If[k >= m/2 + .5, Print[ Prime[n]]], {n, 1, PrimePi[10^6]} ]
%t Select[Range[10^3], Function[n, AllTrue[Binomial[n, Range@ n], SquareFreeQ]]] (* _Michael De Vlieger_, Apr 01 2016, Version 10 *)
%o (PARI) is(n)=for(k=0,n\2,if(!issquarefree(binomial(n,k)),return(0))); 1 \\ _Charles R Greathouse IV_, Mar 06 2014
%Y Cf. A005117, A046098, A048276, A048277, A048279, A249441.
%Y Cf. A000225, A055010, A249452.
%Y Cf. A019532.
%K nonn,fini,full
%O 1,2
%A _Labos Elemer_
%E Edited by _Ralf Stephan_, Aug 03 2004
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