A249717 -id:A249717 - cp's OEIS frontend

A249739 The smallest prime whose square divides n, 1 if n is squarefree.

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, 2, 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, 2, 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, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

A249740 gives the corresponding largest prime.

If n belongs to A013929, then a(n)>1. - Robert G. Wilson v, Nov 16 2016

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

Cf. A003557, A005117, A013929, A020639, A046027, A249717.

Differs from A071773 and A249740 for the first time at n=36, where a(36) = 2, while A249740(36) = 3 and A071773(36) = 6.

Table[If[SquareFreeQ@ n, 1, p = 2; While[! Divisible[n, p^2], p = NextPrime@ p]; p], {n, 120}] (* Michael De Vlieger, Nov 15 2016 *)

a(n) = {f = factor(n/core(n)); vsq = select(x->((x%2) == 0), f[,2], 1); if (#vsq, f[vsq[1], 1], 1);} \\ Michel Marcus, Mar 11 2017

(define (A249739 n) (let loop ((n n) (p (A020639 n))) (cond ((= 1 n) n) ((zero? (modulo n (* p p))) p) (else (loop (/ n p) (A020639 (/ n p)))))))

a(n) = A020639(A003557(n)). - Amiram Eldar, Feb 11 2021

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

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)).

A249718 The largest 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, 3, 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, 3, 3, 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, 5, 5, 5, 5, 2, 2, 2, 5, 3, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2

Offset: 0

Antti Karttunen, Nov 04 2014

nonn

Cf. A249716, A249740.

Differs from A249717 for the first time at n=36, where a(36) = 3, while A249717(36) = 2.

(define (A249718 n) (A249740 (A249716 n)))

a(n) = A249740(A249716(n)).

A249724 Numbers k such that on row k of Pascal's triangle there is no multiple of 9 which would be less than any (potential) multiple of 4 on the same row.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 14, 16, 17, 20, 22, 23, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 48, 49, 50, 51, 52, 53, 56, 57, 58, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 80, 84, 86, 88, 89, 92, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106, 107, 108, 110, 112, 113, 114, 115, 116, 120, 121

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

Disjoint union of {0} and the following sequences: A048278 (gives 7 other cases where there are neither multiples of 4 nor 9 on row k), A249722 (rows where a multiple of 4 is found before a multiple of 9), A249726 (cases where the least term on row k which is a multiple of 4 is also a multiple of 9, and vice versa, i.e., such a term a multiple of 36).

If A249717(k) < 3 then k is included in this sequence. This is a sufficient but not necessary condition, e.g., A249717(25) = 5, but 25 is also included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..23590

Complement: A249723.

Cf. A007318, A052955, A249441, A249695, A249717, A048278, A048645, A051382, A249722, A249726.

A249724list(upto_n) = { my(i=0, n=0, dont_print=0); while(i

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A249739 The smallest prime whose square divides n, 1 if n is squarefree.

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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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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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A249718 The largest 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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A249724 Numbers k such that on row k of Pascal's triangle there is no multiple of 9 which would be less than any (potential) multiple of 4 on the same row.

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