A249439 -id:A249439 - 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

A249440 Positions of records in A249442.

Original entry on oeis.org

0, 4, 6, 13, 15, 43, 47, 239, 7199, 16559, 21599, 33119, 45359, 60479, 90719, 1179359

Offset: 1

Antti Karttunen and Vladimir Shevelev, Nov 04 2014

a(16) = 1179359 is the largest term less than 7,000,000.

From a(8) = 239 onward the decimal representation of all terms seem to end with '9', which indicates that from then on, one larger numbers (A249149) are multiples of ten.

A249439 gives the corresponding record values.
One less than A249149.
Cf. A249442, A249715.

A249714 Record values in A249695.

Original entry on oeis.org

0, 1, 3, 6, 7, 12, 21, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648

Offset: 1

Charles R Greathouse IV & Antti Karttunen, Nov 04 2014

nonn

For n >= 8 [a(8) = 24], the terms seem to be given by A007283(n-5), i.e. as 3 * 2^(n-5).

A249715 gives the positions where these values occur in A249695 for the first time.

Cf. A007283, A249439.

A249695(n) = { for(p=2,3,for(k=0,floor(n/2),if((0==(binomial(n,k)%(p*p))),return(k)))); return(0); } \\ Unoptimized and straightforward.
A249695(n) = { for(p=2,3, my(o=0); for(k=1, n\2, o+=valuation((n-k+1)/k, p); if(o>1, return(k)))); return(0); } \\ Better to use this. Based on Charles R Greathouse IV's PARI-code for A249441.
prevmax = -1; i = 0; for(n=0, 123456789, if((k=A249695(n)) > prevmax, prevmax = k; i++; write("b249714.txt", i, " ", k); write("b249715.txt", i, " ", n))); \\ Compute both A249714 & A249715 at the same time.

a(n) = A249695(A249715(n)).

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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A249440 Positions of records in A249442.

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A249714 Record values in A249695.

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