A249718 -id:A249718 - 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

A249717 The smallest prime whose square divides the first nonsquarefree number on row n of Pascal's triangle, 1 if all terms on that row are squarefree.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 1, 2, 5, 5, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 7, 5, 5, 2, 5, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 5, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 7, 3, 2, 5, 2, 5, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2

Offset: 0

Antti Karttunen, Nov 04 2014

nonn

All such n, for which a(n) < 3, form a subsequence of A249724.

Cf. A249716, A249739, A249724.

Differs from A249718 for the first time at n=36, where a(36) = 2, while A249718(36) = 3.

(define (A249717 n) (A249739 (A249716 n)))

a(n) = A249739(A249716(n)).

A249740 The largest prime whose square divides n, 1 if n is squarefree.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

A249739 gives the corresponding smallest prime.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003557, A005117, A013929, A006530, A052126, A070003, A241917, A249718.

Differs from A071773 and A249739 for the first time at n=36, where a(36) = 3, while A249739(36) = 2 and A071773(36) = 6.

a[n_] := If[(f = Select[FactorInteger[n], Last[#] > 1 &]) == {}, 1, f[[-1, 1]]]; Array[a, 100] (* Amiram Eldar, Feb 11 2021 *)
Table[If[SquareFreeQ[n],1,Select[FactorInteger[n],#[[2]]>1&][[-1,1]]],{n,120}] (* Harvey P. Dale, Feb 28 2021 *)

(define (A249740 n) (let loop ((n n) (p (A006530 n))) (cond ((= 1 n) n) ((zero? (modulo n (* p p))) p) (else (loop (/ n p) (A006530 (/ n p)))))))
;; Alternative version which is based on the given recurrence, and utilizes the memoizing definec-macro from Antti Karttunen's IntSeq-library:
(definec (A249740 n) (cond ((= n 1) n) ((zero? (A241917 n)) (A006530 n)) (else (A249740 (A052126 n)))))

a(1) = 1, and for n > 1, if A241917(n) = 0 [i.e., n is a term of A070003], a(n) = A006530(n), otherwise a(n) = a(A052126(n)).

a(n) = A006530(A003557(n)). - Amiram Eldar, Feb 11 2021

A249716 The least nonsquarefree number on row n of Pascal's triangle, or 1 if all the terms on that row are squarefree.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 20, 1, 8, 9, 45, 1, 12, 1287, 364, 6435, 16, 136, 18, 171, 20, 5985, 1540, 1, 24, 25, 325, 27, 28, 3654, 4060, 169911, 32, 528, 5984, 52360, 36, 666, 8436, 82251, 40, 820, 11480, 145008513, 44, 45, 1035, 12551759587422, 48, 49, 50, 1275, 52, 292825, 54, 1485, 56, 1596, 30856, 45057474, 60, 55525372, 37820, 63, 64, 2080

Offset: 0

Antti Karttunen, Nov 04 2014

nonn

After a(0) = 1, A048278 gives the positions of seven other ones in the sequence.

			           Binomial coefficients     First squarefree     a(n)
                 A007318             occurs at index?      =
----------------------------------------------------------------------------
Row 0                1               no squarefrees        1 (by definition)
Row 1              1   1             no squarefrees        1
Row 2            1   2   1           no squarefrees        1
Row 3          1   3   3   1         no squarefrees        1
Row 4        1   4   6   4   1              1              4
Row 5      1   5  10  10   5   1     no squarefrees        1
Row 6    1   6  15  20  15   6   1          3             20

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

A249717 and A249718 give the smallest and the largest prime whose square divides these numbers.

Cf. also A007318, A005117, A013929, A048277, A048278, A249442.

A249716(n) = { my(b); for(k=0,n\2,if(0==moebius(b=binomial(n,k)),return(b))); return(1); }
for(n=0, 10000, write("b249716.txt", n, " ", A249716(n)));

(define (A249716 n) (A007318tr n (A249442 n)))

a(n) = binomial(n, A249442(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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A249717 The smallest prime whose square divides the first nonsquarefree number on row n of Pascal's triangle, 1 if all terms on that row are squarefree.

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A249740 The largest prime whose square divides n, 1 if n is squarefree.

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A249716 The least nonsquarefree number on row n of Pascal's triangle, or 1 if all the terms on that row are squarefree.

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