OFFSET
0,5
COMMENTS
Number of nonsquarefree numbers (A013929) on row n of Pascal's triangle (A007318). - Antti Karttunen, Nov 05 2014
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..8192
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)
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Definition corrected by Michel Marcus, Mar 05 2014
STATUS
approved