A048279 -id:A048279 - cp's OEIS frontend

A048278 Positive numbers n such that the numbers binomial(n,k) are squarefree for all k = 0..n.

1, 2, 3, 5, 7, 11, 23

Offset: 1

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 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
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

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

A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [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. 373-388.

Cf. A005117, A046098, A048276, A048277, A048279, A249441.
Cf. A000225, A055010, A249452.
Cf. A019532.

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

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 *)

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

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

Edited by Ralf Stephan, Aug 03 2004

A238892 Index of last squarefree number in the first half of row n of Pascal's triangle.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 4, 0, 4, 5, 2, 4, 6, 6, 0, 8, 0, 9, 8, 9, 10, 11, 4, 5, 10, 0, 10, 11, 2, 10, 0, 1, 16, 3, 0, 4, 18, 18, 4, 9, 10, 18, 20, 12, 18, 20, 0, 0, 0, 1, 16, 21, 18, 10, 0, 21, 22, 23, 28, 29, 30, 9, 0, 1, 2, 3, 32, 33, 6, 35, 0, 9, 32, 10, 36

Offset: 0

T. D. Noe, Mar 06 2014

nonn

The first squarefree binomial coefficient in every row is at position 0. Sequence A048279 lists the rows n for which a(n) = 0.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A048279 (positions of zeros).
Cf. A238337 (number of distinct squarefree numbers in row n).
Cf. A238891 (last squarefree number in the first half of row n).

Table[Position[Binomial[n, Range[0, n/2]], _?(SquareFreeQ[#] &)][[-1,1]] - 1, {n, 0, 100}]

cp's OEIS Frontend

A048278 Positive numbers n such that the numbers binomial(n,k) are squarefree for all k = 0..n.

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