A048277 - OEIS

%I #29 Nov 14 2014 12:37:27

%S 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,

%T 25,18,25,20,31,30,27,28,35,30,25,28,37,30,29,18,29,38,27,6,47,48,49,

%U 48,47,36,51,50,55,52,49,38,53,36,23,56,63,62,61,60,61,54,59,54,71,66,57

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

%C Number of nonsquarefree numbers (A013929) on row n of Pascal's triangle (A007318). - _Antti Karttunen_, Nov 05 2014

%H Antti Karttunen, <a href="/A048277/b048277.txt">Table of n, a(n) for n = 0..8192</a>

%F From _Antti Karttunen_, Nov 05 2014: (Start)

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

%F Also, for all n >= 0:

%F a(n) >= A249732(n).

%F a(n) >= A249733(n).

%F (End)

%e 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.

%e 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.

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

%t 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} ]

%t Table[(1 + n) - Length[Select[Binomial[n, Range[0, n]], SquareFreeQ[#] &]], {n, 0, 100}] (* _Vincenzo Librandi_, Nov 06 2014 *)

%o (PARI) a(n) = sum(k=0, n, !issquarefree(binomial(n, k))); \\ _Michel Marcus_, Mar 05 2014

%o (PARI)

%o A048277(n) = sum(k=0,n\2,((0==moebius(binomial(n,k)))*(if(k<(n/2),2,1))));

%o for(n=0, 8192, write("b048277.txt", n, " ", A048277(n))); \\ b-file was computed with this program. - _Antti Karttunen_, Nov 05 2014

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

%K nonn

%O 0,5

%A _Labos Elemer_

%E Definition corrected by _Michel Marcus_, Mar 05 2014