A048277 Number of (not necessarily distinct) nonsquarefree numbers among C(n,k), k=0..n.

0, 0, 0, 0, 2, 0, 1, 0, 6, 8, 5, 0, 9, 4, 3, 2, 15, 12, 17, 12, 13, 12, 11, 0, 21, 22, 19, 26, 25, 18, 25, 20, 31, 30, 27, 28, 35, 30, 25, 28, 37, 30, 29, 18, 29, 38, 27, 6, 47, 48, 49, 48, 47, 36, 51, 50, 55, 52, 49, 38, 53, 36, 23, 56, 63, 62, 61, 60, 61, 54, 59, 54, 71, 66, 57

Offset: 0

Author

Labos Elemer

Keywords

• nonn

Comments

Number of nonsquarefree numbers (A013929) on row n of Pascal's triangle (A007318). - Antti Karttunen, Nov 05 2014

Examples

			a(13) = 4 because C(13,5) = C(13,8) = 3^2*11*13 and C(13,6) = C(13,7) = 2^2*3*11*13.
If n=20, then C[ 20, k ] is divisible by a square for 13 values of k, i.e. for k = 1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, so a[ 20 ] = 13.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192

Crossrefs

Cf. A005117, A007318, A013929, A046098, A048276, A064460, A249732, A249733, A249442, A249716.

Programs

Maple

seq(nops(remove(numtheory:-issqrfree,[seq(binomial(n,k),k=0..n)])),n=0..100); # Robert Israel, Nov 05 2014

Mathematica

f[ n_ ] := (c = 0; k = 1; While[ k < n, If[ Union[ Transpose[ FactorInteger[ Binomial[ n, k ] ] ] [ [ 2 ] ] ] [ [ -1 ] ] > 1, c++ ]; k++ ]; c); Table[ f[ n ], {n, 0, 75} ]
Table[(1 + n) - Length[Select[Binomial[n, Range[0, n]], SquareFreeQ[#] &]], {n, 0, 100}] (* Vincenzo Librandi, Nov 06 2014 *)

PARI

a(n) = sum(k=0, n, !issquarefree(binomial(n, k))); \\ Michel Marcus, Mar 05 2014

PARI

A048277(n) = sum(k=0,n\2,((0==moebius(binomial(n,k)))*(if(k<(n/2),2,1))));
for(n=0, 8192, write("b048277.txt", n, " ", A048277(n))); \\ b-file was computed with this program. - Antti Karttunen, Nov 05 2014

Formula

From Antti Karttunen, Nov 05 2014: (Start)

a(n) = 1 + n - A048276(n).

Also, for all n >= 0:

a(n) >= A249732(n).

a(n) >= A249733(n).

(End)

Extensions

Definition corrected by Michel Marcus, Mar 05 2014