A238337 -id:A238337 - cp's OEIS frontend

A048276 a(n) = number of squarefree numbers among C(n,k), k=0..n.

1, 2, 3, 4, 3, 6, 6, 8, 3, 2, 6, 12, 4, 10, 12, 14, 2, 6, 2, 8, 8, 10, 12, 24, 4, 4, 8, 2, 4, 12, 6, 12, 2, 4, 8, 8, 2, 8, 14, 12, 4, 12, 14, 26, 16, 8, 20, 42, 2, 2, 2, 4, 6, 18, 4, 6, 2, 6, 10, 22, 8, 26, 40, 8, 2, 4, 6, 8, 8, 16, 12, 18, 2, 8, 18, 4, 6, 14, 18, 34, 2, 2, 4, 6, 4, 10, 12, 16, 4

Offset: 0

Labos Elemer

nonn

The only odd numbers are at n = 0, 2, 4, and 8. So this sequence is essentially twice A238337. - T. D. Noe, Mar 07 2014

			If n=20, then C(20, k) is squarefree for k = 0,2,4,8,12,16,18,20, that is, for 8 cases of k, so a(20)=8.

T. D. Noe, Table of n, a(n) for n = 0..5000

Cf. A005117, A046098, A048277, A238337.

A048276 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 0 to n do
        if issqrfree(binomial(n,k)) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A048276(n),n=0..40) ; # R. J. Mathar, Jan 18 2018

Table[Length[Select[Binomial[n, Range[0, n]], SquareFreeQ[#] &]], {n, 0, 100}]

a(n) = sum(k=0, n, issquarefree(binomial(n, k))); \\ Michel Marcus, Dec 19 2013

a(n) = n+1-A048277(n). - R. J. Mathar, Jan 18 2018

A064460 Number of distinct nonsquarefree entries in n-th row of Pascal's triangle.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 3, 4, 3, 0, 5, 2, 2, 1, 8, 6, 9, 6, 7, 6, 6, 0, 11, 11, 10, 13, 13, 9, 13, 10, 16, 15, 14, 14, 18, 15, 13, 14, 19, 15, 15, 9, 15, 19, 14, 3, 24, 24, 25, 24, 24, 18, 26, 25, 28, 26, 25, 19, 27, 18, 12, 28, 32, 31, 31, 30, 31, 27, 30, 27, 36

Offset: 0

Robert G. Wilson v, Oct 02 2001

			a(13) = 2 because C(13,5) = 3^2*11*13 and C(13,6) = 2^2*3*11*13.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A048277, A064461, A064462.

f[ n_ ] := (c = 0; k = 1; While[ k < n/2 + .5, If[ Union[ Transpose[ FactorInteger[ Binomial[ n, k ] ] ] [ [ 2 ] ] ] [ [ -1 ] ] > 1, c++ ]; k++ ]; c); Table[ f[ n ], {n, 0, 100} ]

a(n) = sum(k=0, n\2, !issquarefree(binomial(n, k))); \\ Michel Marcus, Mar 05 2014

a(n) + A238337(n) = A008619(n). - R. J. Mathar, Jan 18 2018

A238891 Largest squarefree number in row n of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 15, 35, 70, 1, 210, 462, 66, 715, 3003, 5005, 1, 24310, 1, 92378, 125970, 293930, 646646, 1352078, 10626, 53130, 5311735, 1, 13123110, 34597290, 435, 44352165, 1, 33, 2203961430, 6545, 1, 66045, 33578000610, 62359143990, 91390, 350343565

Offset: 0

T. D. Noe, Mar 06 2014

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A238337 (number of distinct squarefree numbers in row n).

Cf. A238892 (index of these numbers in the row).

Table[Select[Binomial[n, Range[0, n/2]], SquareFreeQ[#] &][[-1]], {n, 0, 100}]

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}]

A238336 The first row of Pascal's triangle having exactly n distinct squarefree numbers, or -1 if no such row exists.

Original entry on oeis.org

0, 2, 5, 7, 13, 11, 15, 44, 53, 46, 59, 23, 43, 278, 191, 143, 79, 119, 187, 62, 47, 221, 214, 1643, 159, 238, 95, 473, 314, 3583, 671, 474, 958, 3071, 5719, 215, 1439, 7423, 1663, 447, 223, 3695, 4346, 4318, 12983, 319, 35069, 5983, 5471, 8567, 959, 3067

Offset: 1

T. D. Noe, Mar 05 2014

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

Cf. A064461, A238337.

nn = 20; t = Table[-1, {nn}]; found = 0; n = -1; While[found < nn, n++; len = Length[Select[Binomial[n, Range[0, n/2]], SquareFreeQ[#] &]]; If[0 < len <= nn && t[[len]] == -1, t[[len]] = n; found++]]; t

Extended by T. D. Noe, Mar 07 2014

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A048276 a(n) = number of squarefree numbers among C(n,k), k=0..n.

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A064460 Number of distinct nonsquarefree entries in n-th row of Pascal's triangle.

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A238891 Largest squarefree number in row n of Pascal's triangle.

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

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A238336 The first row of Pascal's triangle having exactly n distinct squarefree numbers, or -1 if no such row exists.

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