cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A048278 Positive numbers n such that the numbers binomial(n,k) are squarefree for all k = 0..n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 23
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

It has been shown by Granville and Ramaré that the sequence is complete.
These are all the positive integers m that, when m is represented in binary, contain no composites represented in binary as substrings. - Leroy Quet, Oct 30 2008
This is a number-theoretic sequence, so it automatically assumes that n is positive. To quote Granville and Ramaré, "From Theorem 2 it is evident that there are only finitely many rows of Pascal's Triangle in which all of the entries are squarefree. In section 2 we show that this only occurs in rows 1, 2, 3, 5, 7, 11 and 23 (a result proved by Erdős long ago)." - N. J. A. Sloane, Mar 06 2014
See also comment in A249441. - Vladimir Shevelev, Oct 29 2014
This sequence is equivalent to: Positive integers n such that Fibonacci(n+1) divides n!. This comment depends on the finiteness of A019532. - Altug Alkan, Mar 31 2016

Examples

			n=11: C[11,k] = 1, 11, 55, 165, 330, 462, ... are all squarefree (or 1).

Links

Crossrefs

Cf. A005117, A046098, A048276, A048277, A048279, A249441.
Cf. A000225, A055010, A249452.
Cf. A019532.

Programs

  • Maple
    select(n -> andmap(t -> numtheory:-issqrfree(binomial(n,t)),[$1..floor(n/2)]),[$1..100]); # Robert Israel, Oct 29 2014
  • Mathematica
    Do[m = Prime[n]; k = 2; While[k < m/2 + .5 && Union[ Transpose[ FactorInteger[ Binomial[m, k]]] [[2]]] [[ -1]] < 2, k++ ]; If[k >= m/2 + .5, Print[ Prime[n]]], {n, 1, PrimePi[10^6]} ]
Select[Range[10^3], Function[n, AllTrue[Binomial[n, Range@ n], SquareFreeQ]]] (* Michael De Vlieger, Apr 01 2016, Version 10 *)
  • PARI
    is(n)=for(k=0,n\2,if(!issquarefree(binomial(n,k)),return(0))); 1 \\ Charles R Greathouse IV, Mar 06 2014

Formula

Integers n>0 in set difference between union (A000225, A055010) and A249452. - Vladimir Shevelev, Oct 30 2014
a(n) = A018253(n+1) - 1. - Altug Alkan, Apr 26 2016

Extensions

Edited by Ralf Stephan, Aug 03 2004

A048277 Number of (not necessarily distinct) nonsquarefree numbers among C(n,k), k=0..n.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 0, 6, 8, 5, 0, 9, 4, 3, 2, 15, 12, 17, 12, 13, 12, 11, 0, 21, 22, 19, 26, 25, 18, 25, 20, 31, 30, 27, 28, 35, 30, 25, 28, 37, 30, 29, 18, 29, 38, 27, 6, 47, 48, 49, 48, 47, 36, 51, 50, 55, 52, 49, 38, 53, 36, 23, 56, 63, 62, 61, 60, 61, 54, 59, 54, 71, 66, 57
Offset: 0

Views

Author

Labos Elemer

Keywords

Comments

Number of nonsquarefree numbers (A013929) on row n of Pascal's triangle (A007318). - Antti Karttunen, Nov 05 2014

Examples

			a(13) = 4 because C(13,5) = C(13,8) = 3^2*11*13 and C(13,6) = C(13,7) = 2^2*3*11*13.
If n=20, then C[ 20, k ] is divisible by a square for 13 values of k, i.e. for k = 1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, so a[ 20 ] = 13.

Links

Crossrefs

Cf. A005117, A007318, A013929, A046098, A048276, A064460, A249732, A249733, A249442, A249716.

Programs

  • Maple
    seq(nops(remove(numtheory:-issqrfree,[seq(binomial(n,k),k=0..n)])),n=0..100); # Robert Israel, Nov 05 2014
  • Mathematica
    f[ n_ ] := (c = 0; k = 1; While[ k < n, If[ Union[ Transpose[ FactorInteger[ Binomial[ n, k ] ] ] [ [ 2 ] ] ] [ [ -1 ] ] > 1, c++ ]; k++ ]; c); Table[ f[ n ], {n, 0, 75} ]
Table[(1 + n) - Length[Select[Binomial[n, Range[0, n]], SquareFreeQ[#] &]], {n, 0, 100}] (* Vincenzo Librandi, Nov 06 2014 *)
  • PARI
    a(n) = sum(k=0, n, !issquarefree(binomial(n, k))); \\ Michel Marcus, Mar 05 2014
  • PARI
    A048277(n) = sum(k=0,n\2,((0==moebius(binomial(n,k)))*(if(k<(n/2),2,1))));
for(n=0, 8192, write("b048277.txt", n, " ", A048277(n))); \\ b-file was computed with this program. - Antti Karttunen, Nov 05 2014

Formula

From Antti Karttunen, Nov 05 2014: (Start)
a(n) = 1 + n - A048276(n).
Also, for all n >= 0:
a(n) >= A249732(n).
a(n) >= A249733(n).
(End)

Extensions

Definition corrected by Michel Marcus, Mar 05 2014

A238337 Number of distinct squarefree numbers in row n of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 2, 1, 3, 6, 2, 5, 6, 7, 1, 3, 1, 4, 4, 5, 6, 12, 2, 2, 4, 1, 2, 6, 3, 6, 1, 2, 4, 4, 1, 4, 7, 6, 2, 6, 7, 13, 8, 4, 10, 21, 1, 1, 1, 2, 3, 9, 2, 3, 1, 3, 5, 11, 4, 13, 20, 4, 1, 2, 3, 4, 4, 8, 6, 9, 1, 4, 9, 2, 3, 7, 9, 17, 1, 1, 2, 3, 2
Offset: 0

Views

Author

T. D. Noe, Mar 05 2014

Keywords

Examples

			a(10)=3 because in row 10 of A007318 we observe the three squarefree numbers 1, 10 and 210.

Links

Crossrefs

Cf. A048276 (number of squarefree numbers in the entire row), A238336.

Programs

  • Maple
    A238337 := proc(n)
    local sqf ;
    sqf := {} ;
    for k from 0 to n do
        b := binomial(n,k) ;
        if b=1 or numtheory[issqrfree](b) then
            sqf := sqf union { b} ;
        end if;
    end do:
    nops(sqf) ;
end proc:
seq(A238337(n),n=0..10) ; # R. J. Mathar, Mar 06 2014
  • Mathematica
    Table[Length[Select[Binomial[n, Range[0, n/2]], SquareFreeQ[#] &]], {n, 0, 100}]

Formula

a(n) + A064460(n) = A008619(n). - R. J. Mathar, Jan 18 2018

A064461 First row of Pascal's triangle that has n distinct nonsquarefree entries, or -1 if no such row exists.

Original entry on oeis.org

0, 4, 13, 8, 9, 12, 17, 20, 16, 18, 26, 24, 62, 27, 34, 33, 32, -1, 36, 40, -1, 95, -1, 79, 48, 50, 54, 60, 56, 74, 67, 65, 64, 73, -1, 94, 72, 91, 85, 83, 80, 84, 119, 88, -1, 97, 104, 101, 96, 98, 100, -1, 115, -1, 108, 114, 112, 123, 122, 120, 121, 125, 131
Offset: 0

Views

Author

Robert G. Wilson v, Oct 02 2001

Keywords

Comments

Numbers such that a(n) is -1: 17, 20, 22, 34, 44, 51, ... - Michel Marcus, Mar 05 2014

Examples

			a(2) = 13 because C(13,5) = 3^2*11*13 and C(13,6) = 2^2*3*11*13.

Crossrefs

Cf. A064460, A064462, A048276.

Programs

  • Mathematica
    f[ n_ ] := (c = 0; k = 1; While[ k < n/2 + .5, If[ Union[ Transpose[ FactorInteger[ Binomial[ n, k ] ] ] [ [ 2 ] ] ] [ [ -1 ] ] > 1, c++ ]; k++ ]; c); Do[ m = 2; While[ f[ m ] != n, m++ ]; Print[ m ], {n, 0, 16} ]
  • PARI
    a(n, v) = {for (i=1, #v, if (v[i] == n, return (i-1));); return (-1);} \\ where v is vector A064460; Michel Marcus, Mar 05 2014

Extensions

Corrected and extended by Michel Marcus, Mar 05 2014

A048279 Values of n for which no values of C(n,k) except k=0 and k=n are squarefree.

Original entry on oeis.org

0, 1, 9, 16, 18, 27, 32, 36, 48, 49, 50, 56, 64, 72, 80, 81, 96, 98, 99, 100, 108, 112, 121, 126, 128, 135, 136, 144, 147, 148, 153, 162, 169, 171, 175, 176, 180, 192, 196, 198, 200, 216, 225, 243, 244, 245, 248, 250, 252, 256, 264, 272, 276, 288, 289, 294, 300
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			n=9, C(n,k) = (1),9,36,84,126,126,84,36,9,(1); all values include squares.

Links

Crossrefs

Cf. A048276, A048277, A048278.

Programs

  • Mathematica
    f[n_] := (c = 0; k = 1; While[k < n, If[ Union[ Transpose[ FactorInteger[ Binomial[n, k]]] [[2]]] [[ -1]] > 1, c++ ]; k++ ]; c); Select[Range[150], f[ # ] == # - 1 &]
a[ n_] := If[ n < 2, 0, Module[ {c = 1, k = 1}, While[ c < n, If[ {} == Select[ Table[ Binomial[k, i], {i, k - 1}], SquareFreeQ], c++]; k++]; k - 1]]; (* Michael Somos, Mar 07 2014 *)
Join[{0,1},Select[Range[3,300],NoneTrue[Binomial[#,Range[2,#-1]], SquareFreeQ] &]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 29 2019 *)

Extensions

More terms from Robert G. Wilson v, Oct 02 2001
Showing 1-5 of 5 results.

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