A056057 -id:A056057 - 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

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))).

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

A056175 Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 2, 2, 3, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Jul 27 2000

nonn

Number of prime divisors of the largest square dividing A001405(n). (A prime divisor is nonunitary iff its exponent exceeds 1.)

			For n=10, binomial(10, 5) = 252 = 2*2*3*3*7 has 3 prime divisors of which only one, p=7, is unitary, while 2 and 3 are not. So a(10)=2.
For n=256, binomial(256, 128) also has only 2 prime divisors (3 and 13) whose exponents exceed 1 (4 and 2, respectively), thus a(256)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001405, A034444, A034973, A039593, A056057, A056173.

Table[Count[FactorInteger[Binomial[n, Floor[n/2]]][[All, -1]], e_ /; e > 1], {n, 105}] (* Michael De Vlieger, Mar 05 2017 *)

a(n)=omega(core(binomial(n, n\2), 1)[2]) \\ Charles R Greathouse IV, Mar 09 2017

a(n) = A001221(A000188(A001405(n))).

a(n) = A001221(A056057(n)).

Edited by Jon E. Schoenfield, Mar 05 2017

A056647 a(n) = A056623(A001405(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 1, 9, 9, 1, 4, 1, 4, 4, 1, 1, 4, 100, 25, 100, 25, 9, 144, 9, 9, 1, 4, 25, 100, 100, 25, 9, 36, 4, 1, 4, 1, 25, 400, 225, 900, 1764, 441, 196, 49, 49, 784, 4, 1, 1, 16, 1, 16, 16, 4, 441, 441, 49, 196, 49, 196, 36, 9, 1, 4, 4, 1, 100, 25, 1225

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name "Largest unitary square divisor of central binomial coefficient" was incorrect. See A376553 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056623(binomial(28,14)) = A056623(40116600) = 25.

Cf. A001405, A008833, A055229, A056056, A056057, A056059, A056623, A376553.

a(n) = A008833(A001405(n))/A055229(A001405(n))^2 = A056057(n)/A056059(n)^2.

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

A056670 Largest non-unitary prime factor of A001405(n) = binomial(n, floor(n/2)), or 1 if no such prime exists.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 10 2000

nonn

The largest prime divisor of A056057(n), the largest square divisor of binomial(n, floor(n/2)), or 1 if no such prime exists.

			For n = 28: binomial(28,14) = 2*2*2*3*3*3*5*5*17*19*23, so a(28) = 5.
For n = 342: binomial(342,171) = 32*F, where F is squarefree, so a(341) = 2.

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

Cf. A001405, A006530, A056057, A056175.

a[n_] := Module[{f = Select[FactorInteger[Binomial[n, Floor[n/2]]], Last[#] > 1 &]}, If[f == {}, 1, f[[-1, 1]]]]; Array[a, 100] (* Amiram Eldar, Oct 05 2024 *)

a(n) = A006530(A056057(n)).

A056648 a(n) = A034444(A056647(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 4, 2, 4, 2, 2, 4, 2, 2, 1, 2, 2, 4, 4, 2, 2, 4, 2, 1, 2, 1, 2, 4, 4, 8, 8, 4, 4, 2, 2, 4, 2, 1, 1, 2, 1, 2, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 1, 2, 2, 1, 4, 2, 4, 8, 2, 4, 8, 4, 2, 1, 2, 4, 8, 4, 2, 4, 4, 8, 4, 4, 8, 16, 8, 4, 4, 2, 1, 2, 2, 1

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name, "Number of unitary square divisors of central binomial coefficient", was incorrect. See A376555 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A034444(A056647(28)) = A034444(25) = 2.

Cf. A000188, A001221, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061, A376555.

A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[2^(PrimeNu[ Sqrt[A008833[Binomial[n, Floor[n/2]]]]/A055229[Binomial[n, Floor[n/2]]]]), {n, 1, 25}] (* G. C. Greubel, May 20 2017 *)

a(n) = 2^r, where r = A001221(A000188(A001405(n))/A055229(A001405(n))).

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

A056649 a(n) = A056061(n) - A034444(A056647(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 4, 6, 2, 2, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 6, 8, 0, 0, 0, 4, 4, 6, 2, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 4, 4, 0, 0, 4, 8, 2, 3, 6, 8, 4, 8, 2, 2, 4, 4, 8, 8, 0, 0, 0, 4, 2, 4, 3, 4, 2, 3, 4

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name, "Number of non-unitary square divisors of central binomial coefficient", was incorrect. See A376556 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056061(28) - A034444(A056647(28)) = A056061(28) - A034444(25) = 8 - 2 = 6.

Cf. A000188, A001221, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061, A376556.

A056061[n_] := Count[Divisors@Binomial[n, Floor[n/2]], d_ /; IntegerQ@Sqrt@d]; A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]];
Table[A056061[n] - 2^(PrimeNu[Sqrt[A008833[Binomial[n, Floor[n/2]]]]/ A055229[Binomial[n, Floor[n/2]]]]), {n, 1, 15}] (* G. C. Greubel, May 20 2017 *)

a(n) = A056061(n) - 2^r, where r = A001221(A000188(A001405(n))/A055229(A001405(n))).

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

cp's OEIS Frontend

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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A056061 Number of square divisors of central binomial coefficients.

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A056060 The powerfree part of the central binomial coefficients.

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A056175 Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

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A056647 a(n) = A056623(A001405(n)).

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A056670 Largest non-unitary prime factor of A001405(n) = binomial(n, floor(n/2)), or 1 if no such prime exists.

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A056648 a(n) = A034444(A056647(n)).