A046098 -id:A046098 - cp's OEIS frontend

A064033 Product of non-unitary divisors of binomial(n, floor(n/2)) or a(n) = 1 if all divisors are unitary. See A046098.

1, 1, 1, 1, 1, 20, 1, 1, 15876, 1016255020032, 1, 728933458176, 8670998958336, 19247673071478783248355557376, 1714723915100625, 752711194884611945703392100000000, 1, 31226235883841773375939805209600000000, 1, 1357651828905889565182743230460164655087616

Offset: 1

Labos Elemer, Sep 13 2001

nonn

Cf. A001405, A048243, A056173, A061537, A061538.

f[n_] := n^((DivisorSigma[0, n] - 2^PrimeNu[n]) / 2); Table[f[Binomial[n, Floor[n/2]]], {n, 1, 20}] (* Amiram Eldar, Jul 22 2024 *)

a(n) = apply(x -> x^((numdiv(x) - 2^omega(x))/2), binomial(n, n\2)); \\ Amiram Eldar, Jul 22 2024

a(n) = A061538(A001405(n)).

a(18)-a(20) from Amiram Eldar, Jul 22 2024

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

Labos Elemer

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

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

A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI].
H. N. Ramaswamy and R. Siddaramu, On the Stufe, unit Stufe and Pythagoras number of the ring of integers modulo n, Adv. Stud. Contemp. Math. (Kyungshang) 20:3 (2010), pp. 373-388.

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

select(n -> andmap(t -> numtheory:-issqrfree(binomial(n,t)),[$1..floor(n/2)]),[$1..100]); # Robert Israel, Oct 29 2014

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 *)

is(n)=for(k=0,n\2,if(!issquarefree(binomial(n,k)),return(0))); 1 \\ Charles R Greathouse IV, Mar 06 2014

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

Edited by Ralf Stephan, Aug 03 2004

A056059 GCD of largest square and squarefree part of central binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 6, 2, 1, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 6, 3, 1, 1, 1, 2, 3, 6, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 3, 6, 6, 3, 1, 2, 2, 1, 2, 1, 3, 6, 1, 1, 1, 2, 1, 2

Offset: 1

Labos Elemer, Jul 26 2000

nonn

			n=14, binomial(14,7) = 3432 = 8*3*11*13. The largest square divisor is 4, squarefree part is 858. So a(14) = gcd(4,858) = 2.

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

Cf. A000188, A001405, A007913, A008833, A034974, A046098, A055229, A055231.
Cf. A056056, A056057, A056058, A056060, A056061.
A056201 is the cube of this sequence.

Table[GCD[First@ Select[Reverse@ Divisors@ #, IntegerQ@ Sqrt@ # &], Times @@ Power @@@ Map[{#1, Mod[#2, 2]} & @@ # &, FactorInteger@ #]] &@ Binomial[n, Floor[n/2]], {n, 80}] (* Michael De Vlieger, Feb 18 2017, after Zak Seidov at A007913 *)

A001405(n) = binomial(n, n\2);
A055229(n) = { my(c=core(n)); gcd(c, n/c); } \\ Charles R Greathouse IV, Nov 20 2012
A056059(n) = A055229(A001405(n)); \\ Antti Karttunen, Jul 20 2017

from sympy import binomial, gcd
from sympy.ntheory.factor_ import core
def a001405(n): return binomial(n, n//2)
def a055229(n):
    c = core(n)
    return gcd(c, n//c)
def a(n): return a055229(a001405(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Jul 20 2017

a(n) = A055229(A001405(n)), where A055229(n) = gcd(A008833(n), A007913(n)).

Formula clarified by Antti Karttunen, Jul 20 2017

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

Labos Elemer

nonn

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

			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.

Antti Karttunen, Table of n, a(n) for n = 0..8192

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

seq(nops(remove(numtheory:-issqrfree,[seq(binomial(n,k),k=0..n)])),n=0..100); # Robert Israel, Nov 05 2014

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 *)

a(n) = sum(k=0, n, !issquarefree(binomial(n, k))); \\ Michel Marcus, Mar 05 2014

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

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)

Definition corrected by Michel Marcus, Mar 05 2014

A056057 The largest square which divides n-th central binomial coefficient.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 4, 9, 9, 1, 4, 1, 4, 4, 4, 1, 4, 100, 100, 900, 900, 36, 144, 9, 9, 9, 36, 25, 100, 100, 100, 9, 36, 4, 4, 4, 4, 900, 3600, 225, 900, 1764, 1764, 1764, 1764, 196, 784, 4, 4, 4, 16, 4, 16, 16, 16, 441, 441, 49, 196, 49, 196, 36, 36, 1, 4

Offset: 1

Labos Elemer, Jul 26 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974.

Cf. A056056, A056058, A056059, A056060, A056061.

Table[First@ Select[Reverse@ Divisors@ Binomial[n, Floor[n/2]], IntegerQ@ Sqrt@ # &], {n, 72}] (* Michael De Vlieger, Feb 18 2017 *)
a[n_] := Times @@ (First[#]^(2*Floor[Last[#]/2]) & /@ FactorInteger[Binomial[n, Floor[n/2]]]); Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

a(n) = A008833(A001405(n)).

a(A046098(n)) = 1.

A056060 The powerfree part of the central binomial coefficients.

Original entry on oeis.org

1, 2, 3, 6, 10, 5, 35, 70, 14, 7, 462, 231, 429, 429, 715, 1430, 24310, 12155, 92378, 46189, 88179, 88179, 1352078, 676039, 52003, 52003, 7429, 7429, 1077205, 1077205, 33393355, 66786710, 43214930, 21607465, 181502706, 90751353, 176726319, 176726319, 7658140490

Offset: 1

Labos Elemer, Jul 26 2000

nonn

			n=14, binomial(14,7) = 3432 = 8*3*11*13. The largest square divisor is 4, and the squarefree part is 858. So GCD(4,858) = 2 and a(14) = 858/2 = 429.

Amiram Eldar, Table of n, a(n) for n = 1..3388

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974, A056056, A056057, A056058, A056059, A056060, A056061.

a[n_] := Denominator[(b = Binomial[n, Floor[n/2]])/(Times @@ First /@ FactorInteger[b])^2]; Array[a, 36] (* Amiram Eldar, Sep 05 2020 *)

a(n) = A055231(A001405(n)).

New name and more terms from Amiram Eldar, Sep 05 2020

A056058 Squarefree part of the n-th central binomial coefficient.

Original entry on oeis.org

1, 2, 3, 6, 10, 5, 35, 70, 14, 7, 462, 231, 429, 858, 715, 1430, 24310, 12155, 92378, 46189, 88179, 176358, 1352078, 676039, 52003, 104006, 22287, 44574, 2154410, 1077205, 33393355, 66786710, 129644790, 64822395, 181502706, 90751353

Offset: 1

Labos Elemer, Jul 26 2000

nonn

a(3387) has 1001 decimal digits. - Michael De Vlieger, Jul 14 2017

Michael De Vlieger, Table of n, a(n) for n = 1..3386

Cf. A048633 for another version.

Table[Sqrt[Binomial[n, Floor[n/2]]] /. (c_: 1) a_^(b_: 0) :> (c a^b)^2, {n, 36}] (* Michael De Vlieger, Jul 14 2017, after Bill Gosper at A007913 *)

a(n) = A007913(A001405(n)).

For squarefree central binomial coefficients (A046098), a(n)=A001405(n).

A048276 a(n) = number of squarefree numbers among C(n,k), k=0..n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 6, 8, 3, 2, 6, 12, 4, 10, 12, 14, 2, 6, 2, 8, 8, 10, 12, 24, 4, 4, 8, 2, 4, 12, 6, 12, 2, 4, 8, 8, 2, 8, 14, 12, 4, 12, 14, 26, 16, 8, 20, 42, 2, 2, 2, 4, 6, 18, 4, 6, 2, 6, 10, 22, 8, 26, 40, 8, 2, 4, 6, 8, 8, 16, 12, 18, 2, 8, 18, 4, 6, 14, 18, 34, 2, 2, 4, 6, 4, 10, 12, 16, 4

Offset: 0

Labos Elemer

nonn

The only odd numbers are at n = 0, 2, 4, and 8. So this sequence is essentially twice A238337. - T. D. Noe, Mar 07 2014

			If n=20, then C(20, k) is squarefree for k = 0,2,4,8,12,16,18,20, that is, for 8 cases of k, so a(20)=8.

T. D. Noe, Table of n, a(n) for n = 0..5000

Cf. A005117, A046098, A048277, A238337.

A048276 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 0 to n do
        if issqrfree(binomial(n,k)) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A048276(n),n=0..40) ; # R. J. Mathar, Jan 18 2018

Table[Length[Select[Binomial[n, Range[0, n]], SquareFreeQ[#] &]], {n, 0, 100}]

a(n) = sum(k=0, n, issquarefree(binomial(n, k))); \\ Michel Marcus, Dec 19 2013

a(n) = n+1-A048277(n). - R. J. Mathar, Jan 18 2018

A056651 Numbers k such that binomial(k,floor(k/2)) has no non-unitary square divisors: all of their square divisors are unitary ones.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 31, 32, 35, 36, 37, 39, 40, 41, 43, 47, 48, 49, 55, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 75, 79, 80, 95, 96, 97, 111, 129, 130, 131, 132, 133, 143, 144, 151, 161, 163, 167, 191, 192, 193

Offset: 1

Labos Elemer, Aug 09 2000

nonn

This property is weaker than "squarefreedom", but shows how central binomial coefficients are "poor of squares".

Numbers k such that binomial(k,floor(k/2)) is cubefree (A004709). - Amiram Eldar, Jul 22 2024

			223 is a term because x = binomial(223,111) has 35 prime divisors. 33 arises at power 1. Only 2 and 13 has powers 2 > 1. So square divisors of x are {1, 4, 169, 676} ={s}. All of them are also unitary divisors since GCD(s,x/s) = 1 holds for them.

T. D. Noe, Table of n, a(n) for n = 1..129 (no others < 10^8)

Cf. A001405, A004709, A046098, A056175, A110495.

Select[Range[0, 11000], AllTrue[FactorInteger[Binomial[#, Floor[#/2]]][[;;, 2]], #1 <= 2 &] &] (* Amiram Eldar, Jul 22 2024 *)

is(n) = if(n <= 1, 1, vecmax(factor(binomial(n, floor(n/2)))[, 2]) < 3); \\ Amiram Eldar, Jul 22 2024

A081391 Numbers k such that the central binomial coefficient C(2*k,k) has only one prime divisor whose exponent equals one.

Original entry on oeis.org

3, 6, 7, 8, 9, 10, 11, 12, 16, 21, 22, 28, 29, 30, 31, 36, 37, 54, 55, 57, 58, 110, 171, 784, 786

Offset: 1

Labos Elemer, Mar 27 2003

Numbers k such that C(2*k,k) has one non-unitary prime divisor.
Numbers k for which A081387(k) = 1.
No more terms through 10^6; conjecture: no terms after 786. - Jon E. Schoenfield, Jul 29 2017

			For k = 786, C(1572,786) = 2*2*2*2*m, where m is a squarefree product of 169 primes.

Cf. A000108, A000984, A001405, A046098, A081386, A081387, A081388, A081389, A081390, A080664.

q[k_] := Count[FactorInteger[Binomial[2*k, k]][[;;, 2]], ?( # > 1 &)] == 1; Select[Range[1000], q] (* _Amiram Eldar, Oct 05 2024 *)

is(k) = {my(e = factor(binomial(2*k, k))[, 2]); sum(i = 1, #e, e[i] > 1) == 1;} \\ Amiram Eldar, Oct 05 2024

cp's OEIS Frontend

A064033 Product of non-unitary divisors of binomial(n, floor(n/2)) or a(n) = 1 if all divisors are unitary. See A046098.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A056059 GCD of largest square and squarefree part of central binomial coefficients.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A056057 The largest square which divides n-th central binomial coefficient.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A056060 The powerfree part of the central binomial coefficients.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A056058 Squarefree part of the n-th central binomial coefficient.

Views

Author

Keywords

Comments

Links

Crossrefs