

A048278


Positive numbers n such that the numbers binomial(n,k) are squarefree for all k = 0..n.


13




OFFSET

1,2


COMMENTS

It has been shown by Granville and Ramaré that the sequence is complete.
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
This is a numbertheoretic 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
See also comment in A249441.  Vladimir Shevelev, Oct 29 2014
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


LINKS

Table of n, a(n) for n=1..7.
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73107, [DOI].
H. N. Ramaswamy and R. Siddaramu, On the Stufe, unit Stufe and Pythagoras number of the ring of integers modulo n, Adv. Stud. Contemp. Math. (Kyungshang) 20:3 (2010), pp. 373388.


FORMULA

Integers n>0 in set difference between union (A000225, A055010) and A249452.  Vladimir Shevelev, Oct 30 2014
a(n) = A018253(n+1)  1.  Altug Alkan, Apr 26 2016


EXAMPLE

n=11: C[11,k] = 1, 11, 55, 165, 330, 462, ... are all squarefree (or 1).


MAPLE

select(n > andmap(t > numtheory:issqrfree(binomial(n, t)), [$1..floor(n/2)]), [$1..100]); # Robert Israel, Oct 29 2014


MATHEMATICA

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]} ]
Select[Range[10^3], Function[n, AllTrue[Binomial[n, Range@ n], SquareFreeQ]]] (* Michael De Vlieger, Apr 01 2016, Version 10 *)


PROG

(PARI) is(n)=for(k=0, n\2, if(!issquarefree(binomial(n, k)), return(0))); 1 \\ Charles R Greathouse IV, Mar 06 2014


CROSSREFS

Cf. A005117, A046098, A048276, A048277, A048279, A249441.
Cf. A000225, A055010, A249452.
Cf. A019532.
Sequence in context: A061165 A183055 A046689 * A247797 A068863 A284146
Adjacent sequences: A048275 A048276 A048277 * A048279 A048280 A048281


KEYWORD

nonn,fini,full


AUTHOR

Labos Elemer


EXTENSIONS

Edited by Ralf Stephan, Aug 03 2004


STATUS

approved



