A103447 -id:A103447 - cp's OEIS frontend

A105594 Triangle read by rows: abs(A103447)*A047999 mod 2.

1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1

Offset: 0

Paul Barry, Apr 14 2005

Row sums are A105595.

			Triangle starts
1;
0,1;
1,1,1;
0,0,0,1;
1,0,1,0,1;
0,1,0,1,0,1;
0,0,0,0,1,1,1;

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Cf. A047999, A103447, A105595, A105596.

A105594 := proc(n,k)
    add( abs(numtheory[mobius](binomial(n,j)))*modp(binomial(j,k),2) ,j=0..n) ;
    % mod 2 ;
end proc: # R. J. Mathar, Nov 28 2014

T[n_, k_] := Sum[Abs[MoebiusMu[Binomial[n, j]]*Mod[Binomial[j, k], 2]], {j, 0, n}] // Mod[#, 2]&;
Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 15 2020 *)

T(n, k) = mod(Sum_{j=0..n}(abs(mu(binomial(n,j)))*mod(binomial(j,k),2)), 2).

A249442 a(n) is the smallest m such that binomial(n,m) is not squarefree, or a(n)=0, if there is no such m.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 1, 1, 2, 0, 1, 5, 3, 7, 1, 2, 1, 2, 1, 4, 3, 0, 1, 1, 2, 1, 1, 3, 3, 5, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 8, 1, 1, 2, 21, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 3, 6, 1, 6, 3, 1, 1, 2, 3, 4, 1, 6, 3, 8, 1, 2, 3, 1, 1, 3, 3, 8, 1, 1, 2, 3, 1, 5, 3

Offset: 0

Antti Karttunen and Vladimir Shevelev, Oct 28 2014

nonn

The sequence gives the position of the first zero on row n (both starting from zero) in the triangular table A103447, and zero if there is no zero on that row. After a(0) = 0, A048278 gives the positions of seven other zeros in the sequence.

Records are 0,1,3,5,7,8,21,... (A249439) in positions 0,4,6,13,15,43,47,... (A249440).

Antti Karttunen, Table of n, a(n) for n = 0..10000

A249439 gives the record values, A249440 the positions where they occur for the first time.
Cf. A005117, A007913, A048277, A048278, A103447, A249716, A249717, A249718.
Differs from A249695 for the first time at n=9.

Table[If[#>n,0,#]&[NestWhile[#+1&,1,SquareFreeQ[Binomial[n,#]]&]],{n,0,100}] (* Peter J. C. Moses, Nov 04 2014 *)

A249442(n) = { for(k=0,n\2,if(0==moebius(binomial(n,k)),return(k))); return(0); }
for(n=0, 10000, write("b249442.txt", n, " ", A249442(n)));
\\ Antti Karttunen, Nov 04 2014

Other identities:

A249716(n) = binomial(n, a(n)). [A249716(n) gives the corresponding minimal nonsquarefree binomial coefficient, or 1 when n is one of the terms of A048278].

More terms from Peter J. C. Moses, Oct 28 2014

A103448 a(n) = Sum_{k=0..n} Moebius(binomial(n,k)).

Original entry on oeis.org

1, 2, 1, 0, 3, 2, 6, 4, 1, 2, 6, 4, 0, -6, 8, 6, 2, -2, 2, -4, 4, 10, 4, 8, 0, 4, 8, 2, 4, 0, -2, -4, 2, 4, 0, 4, 2, -4, 10, 4, 0, -8, 6, -2, 4, -4, 8, 2, 2, 2, 2, 4, 6, 2, 0, 6, 2, 2, 2, -6, 0, 6, 4, 8, 2, 4, 2, 0, 0, 8, -4, -2, 2, 4, 2, 0, -2, 14, 10, -2, 2, 2, 4, 2, 4, -2, 0, 8, 4, 2, 2, -2, 6, 0, -6, 14, 2, 0, 2, 2, 2, 4, 0, 2, -2

Offset: 0

Emeric Deutsch, Feb 07 2005

sign

Row sums of A103447.

			a(4)=3 because mu(1) + mu(4) + mu(6) + mu(4) + mu(1) = 1 + 0 + 1 + 0 + 1 = 3.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A007318, A008683, A103447, A103449.

[(&+[ MoebiusMu(Binomial(n, k)): k in [0..n]]): n in [0..120]]; // G. C. Greubel, Jun 16 2021

Table[Sum[MoebiusMu[Binomial[n, k]], {k,0,n}], {n,0,120}] (* G. C. Greubel, Jun 16 2021 *)

a(n) = sum(k=0, n, moebius(binomial(n, k))); \\ Michel Marcus, Jun 17 2021

[sum(moebius(binomial(n, k)) for k in (0..n)) for n in (0..120)] # G. C. Greubel, Jun 16 2021

a(n) = Sum_{k=0..n} Moebius(binomial(n,k)).

a(n) = Sum_{k=0..n} A008683(A007318(n,k)).

A103449 Values of k such that Sum_{m=0..k} Moebius(binomial(k,m)) = 0.

Original entry on oeis.org

3, 12, 24, 29, 34, 40, 54, 60, 67, 68, 75, 86, 93, 97, 102, 119, 125, 131, 133, 142, 152, 157, 160, 163, 164, 168, 170, 172, 189, 193, 197, 208, 210, 220, 221, 228, 229, 246, 251, 255, 257, 261, 270, 275, 280, 293, 296, 307, 308, 313, 315, 332, 337, 338, 340

Offset: 1

Emeric Deutsch, Feb 07 2005

nonn

Values of k such that A103448(k) = 0.

			12 belongs to the sequence because the only squarefree values of binomial(12,m) are 1, 2*3*11, 2*3*11, 1, on which the Mobius function takes the values 1,-1,-1,1, respectively.
8 does not belong to the sequence because the only squarefree value of binomial(8,m) are 1, 2*5*7, 1, on which the Moebius function takes the values 1,-1,1, respectively.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A008683, A103447, A103448.

A103448[n_]:= A103448[n]= Sum[MoebiusMu[Binomial[n, k]], {k, 0, n}];
f:= Table[A103448[n], {n, 0, 1050}];
Select[Range[0, 1000], f[[#]] == 0 &] - 1  (* G. C. Greubel, Jun 17 2021 *)

is(k) = sum(m=0, k, moebius(binomial(k, m)))==0 \\ Felix Fröhlich, Jun 18 2021

cp's OEIS Frontend

A105594 Triangle read by rows: abs(A103447)*A047999 mod 2.

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A249442 a(n) is the smallest m such that binomial(n,m) is not squarefree, or a(n)=0, if there is no such m.

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A103448 a(n) = Sum_{k=0..n} Moebius(binomial(n,k)).

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A103449 Values of k such that Sum_{m=0..k} Moebius(binomial(k,m)) = 0.

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