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 1-7 of 7 results.

A056673 Number of unitary and squarefree divisors of binomial(n, floor(n/2)). Also the number of divisors of the powerfree part of A001405(n), A056060(n).

Original entry on oeis.org

1, 2, 2, 4, 4, 2, 4, 8, 4, 2, 16, 8, 8, 8, 8, 16, 32, 16, 32, 16, 32, 32, 64, 32, 16, 16, 8, 8, 32, 32, 64, 128, 128, 64, 256, 128, 128, 128, 512, 256, 512, 512, 512, 512, 64, 64, 256, 128, 128, 128, 128, 128, 256, 256, 2048, 2048, 4096, 4096, 2048, 2048, 2048, 2048
Offset: 1

Views

Author

Labos Elemer, Aug 10 2000

Keywords

Examples

			n = 14: binomial(15,7) = 3432 = 2*2*2*3*11*13, which has 32 divisors. Of those divisors, 16 are unitary: {1, 3, 8, 11, 13, 24, 33, 39, 88, 104, 143, 264, 312, 429, 1144, 3432}; 16 are squarefree: {1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858}. Only 8 of the divisors belong to both classes: {1, 3, 11, 13, 33, 39, 143, 429}. Thus, a(14) = 8.

Links

Crossrefs

Cf. A000005, A001405, A007913, A055229, A055231, A056060, A056671.

Programs

  • Mathematica
    Table[With[{m = Binomial[n, Floor[n/2]]}, DivisorSum[m, 1 &, And[CoprimeQ[#, m/#], SquareFreeQ@ #] &]], {n, 62}] (* Michael De Vlieger, Sep 05 2017 *)
f[p_, e_] := If[e == 1, 2, 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[ Binomial[n, Floor[n/2]]]); Array[a, 60] (* Amiram Eldar, Sep 06 2020 *)
  • PARI
    a(n) = my(b=binomial(n, n\2)); sumdiv(b, d, issquarefree(d) && (gcd(d, b/d) == 1)); \\ Michel Marcus, Sep 05 2017

Formula

a(n) = A000005(A055231(x)) = A000005(A007913(x)/A055229(x)), where x = A001405(n) = binomial(n, floor(n/2)).
a(n) = A056671(A001405(n)). - Amiram Eldar, Sep 06 2020

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Jul 26 2000

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Labos Elemer, Jul 26 2000

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Python
    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

Formula

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

Extensions

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

Views

Author

Labos Elemer, Jul 26 2000

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Labos Elemer Jul 26 2000

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Count[Divisors@ Binomial[n, Floor[n/2]], d_ /; IntegerQ@ Sqrt@ d], {n, 0, 84}] (* Michael De Vlieger, Feb 18 2017 *)
  • PARI
    a(n) = sumdiv(binomial(n, n\2), d, issquare(d)); \\ Michel Marcus, Feb 19 2017

Formula

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

A056201 Characteristic cube divisor (A056191) of central binomial coefficient (A001405).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 8, 27, 216, 8, 1, 1, 1, 27, 27, 1, 1, 1, 8, 1, 1, 1, 8, 1, 8, 216, 27, 1, 1, 1, 8, 27, 216, 8, 1, 1, 8, 8, 1, 8, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 8, 1, 8, 8, 1, 1, 1, 1, 8, 27, 216, 216, 27, 1, 8, 8, 1, 8, 1, 27, 216
Offset: 1

Views

Author

Labos Elemer, Aug 02 2000

Keywords

Examples

			n=14, binomial(14,8) = 3432 = 2*2*2*3*11*13 so g(3432)=2, thus a(14)=8.

Crossrefs

Cf. A056056-A056060, A056202.
Equals A056059^3.

Formula

a(n) = A056059(n)^3 = g^3 and binomial(n, floor(n/2)) = a(n)^3 * L^2 * A056060(n), where L = A056056(n)/A056059(n).

A056202 Central binomial coefficient A001405(n) divided by its characteristic cube divisor A056201(n).

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 429, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 88179, 1352078, 2704156, 5200300, 1300075, 742900, 185725, 9694845, 155117520, 300540195, 601080390, 43214930, 86429860
Offset: 1

Views

Author

Labos Elemer, Aug 02 2000

Keywords

Examples

			n=14, binomial(14,7) = 3432 and A056059(14) = 2, thus a(14) = 3432/(2*2*2) = 429.

Crossrefs

Cf. A001405, A056056-A056060, A056201.

Formula

a(n) = A001405(n)/A056059(n)^3 = binomial(n, floor(n/2))/A056059(n)^3 = A001405(n)/A056201(n).
Showing 1-7 of 7 results.

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