A249440 -id:A249440 - cp's OEIS frontend

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.

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

A249439 Record values in A249442.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 21, 24, 32, 36, 40, 45, 48, 64, 91, 94

Offset: 1

Antti Karttunen and Vladimir Shevelev, Nov 04 2014

A249440 gives the positions where these values occur in A249442 for the first time.

A249442(n) = { for(k=0,n\2,if(0==moebius(binomial(n,k)),return(k))); return(0); }
prevmax = -1; i = 0; for(n=0, 123456789, if((k=A249442(n)) > prevmax, prevmax = k; i++; write("b249439.txt", i, " ", k); write("b249440.txt", i, " ", n)));
\\ Compute both A249439 and A249440 at the same time. - Antti Karttunen, Nov 04 2014

a(n) = A249442(A249440(n)).

A249715 Positions of records in A249695.

Original entry on oeis.org

0, 4, 6, 13, 15, 27, 47, 55, 111, 223, 447, 895, 1791, 3583, 7167, 14335, 28671, 57343, 114687, 229375, 458751, 917503, 1835007, 3670015, 7340031, 14680063, 29360127, 58720255, 117440511

Offset: 1

Charles R Greathouse IV & Antti Karttunen, Nov 04 2014

nonn

From n=8 [a(8)=55] onward, the terms seem to be given by A086224(n-5), i.e. as (7 * 2^(n-5)) - 1.

A249714 gives the corresponding record values.

Cf. A249695, A086224, A249440.

A249149 One more than positions of records in A249442.

Original entry on oeis.org

1, 5, 7, 14, 16, 44, 48, 240, 7200, 16560, 21600, 33120, 45360, 60480, 90720, 1179360

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

After 1, these terms factorize as: 5, 7, 2 * 7, 2^4, 2^2 * 11, 2^4 * 3, 2^4 * 3 * 5, 2^5 * 3^2 * 5^2, 2^4 * 3^2 * 5 * 23, 2^5 * 3^3 * 5^2, 2^5 * 3^2 * 5 * 23, 2^4 * 3^4 * 5 * 7, 2^6 * 3^3 * 5 * 7, 2^5 * 3^4 * 5 * 7, 2^5 * 3^4 * 5 * 7 * 13, ...

One more than A249440.

a(n) = A249440(n)+1.

cp's OEIS Frontend

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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A249439 Record values in A249442.

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A249715 Positions of records in A249695.

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A249149 One more than positions of records in A249442.

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