A056058 -id:A056058 - cp's OEIS frontend

A056056 Square root of largest square dividing n-th central binomial coefficient.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 2, 2, 2, 3, 3, 1, 2, 1, 2, 2, 2, 1, 2, 10, 10, 30, 30, 6, 12, 3, 3, 3, 6, 5, 10, 10, 10, 3, 6, 2, 2, 2, 2, 30, 60, 15, 30, 42, 42, 42, 42, 14, 28, 2, 2, 2, 4, 2, 4, 4, 4, 21, 21, 7, 14, 7, 14, 6, 6, 1, 2, 2, 2, 10, 10, 70, 140, 7, 14, 126, 126, 6, 6, 30, 60

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, A056057, A056058, A056059, A056060, A056061.

Table[Sqrt@ Max@ Select[Divisors@ Binomial[n, Floor[n/2]], IntegerQ@ Sqrt@ # &], {n, 0, 86}] (* Michael De Vlieger, Jul 04 2016 *)
a[n_] := Times @@ (First[#]^Floor[Last[#]/2] & /@ FactorInteger[Binomial[n, Floor[n/2]]]); Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

a(n) = b = binomial(n, n\2); sqrtint(b/core(b)); \\ Michel Marcus, Dec 10 2013

a(n) = A000188(A001405(n)).

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

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.

A056061 Number of square divisors of central binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 8, 8, 4, 6, 2, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 2, 8, 12, 4, 8, 8, 8, 8, 8, 4, 6, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 2, 4, 2, 4, 4, 4, 1, 2, 2, 2, 4, 4, 8, 12, 2, 4, 12, 12, 4, 4, 8, 12, 12, 12, 4, 6, 8, 12, 12, 12, 8, 16, 8, 8, 6

Offset: 1

Labos Elemer Jul 26 2000

nonn

			n=27: binomial(27,13) = 20058300, its largest square-divisor is 900=30^2 so a(27) = tau(30) = 8.

Michel Marcus, Table of n, a(n) for n = 1..120

Cf. A001405, A046951, A000005, A000188, A056056, A056057, A056058, A056059, A056060.

Table[Count[Divisors@ Binomial[n, Floor[n/2]], d_ /; IntegerQ@ Sqrt@ d], {n, 0, 84}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = sumdiv(binomial(n, n\2), d, issquare(d)); \\ Michel Marcus, Feb 19 2017

a(n) = A046951(A001405(n)) = A000005(A000188(A001405(n))).

A048633 Largest squarefree number dividing n-th central binomial coefficient C(n,[ n/2 ]).

Original entry on oeis.org

1, 2, 3, 6, 10, 10, 35, 70, 42, 42, 462, 462, 858, 858, 2145, 4290, 24310, 24310, 92378, 92378, 176358, 176358, 1352078, 1352078, 520030, 520030, 222870, 222870, 6463230, 6463230, 100180065, 200360130, 129644790, 129644790, 907513530

Offset: 1

Labos Elemer

nonn

a(2k+1)=a(2k+2) unless 2k+1 is in A000225, in which case a(2k+2)=2*a(2k+1). - Robert Israel, Jan 21 2020

			n=10: C(10,5)=252=2*2*3*3*7. The largest squarefree number dividing the 10th central binomial coefficient is 2*3*7=42. Thus a(10)=42

Robert Israel, Table of n, a(n) for n = 1..3364

Equals A007947(A001405(n)). Cf. A034973, A000225.

See A056058 for another version.

[&*PrimeDivisors(Binomial(n, Floor(n/2))): n in [1..35]]; // Marius A. Burtea, Jan 21 2020

f:= n -> convert(numtheory:-factorset(binomial(n,floor(n/2))),`*`):
map(f, [$1..50]); # Robert Israel, Jan 21 2020

Table[Last@ Select[Divisors@ Binomial[n, Floor[n/2]], SquareFreeQ], {n, 35}] (* Michael De Vlieger, Feb 05 2017 *)

a(n)=factorback(factor(binomial(n,n\2))[,1]) \\ Charles R Greathouse IV, Nov 05 2017

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

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A056056 Square root of largest square dividing n-th central binomial coefficient.

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

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

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A056061 Number of square divisors of central binomial coefficients.

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A048633 Largest squarefree number dividing n-th central binomial coefficient C(n,[ n/2 ]).

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A056060 The powerfree part of the central binomial coefficients.

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