A282163 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^3.

1, 154836, 985320, 1108536, 1113959, 1492260, 1576696, 1632708, 1649238, 1684540, 1805570, 1988008, 2508792, 2548810, 2550408, 2659260, 2698740, 2746590, 2995122, 3074552, 3286710, 3330795, 3538458, 3574200, 3730155, 4039932, 4160240, 4318548, 4374370, 4426695, 4523985

Offset

1,2

Comments

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^3. - Lucian Craciun, Feb 09 2017

The asymptotic density of this sequence is 0.000031511777... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

Links

Lucian Craciun, Table of n, a(n) for n = 1..16000 (corrected and extended by Giovanni Resta)

Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.

Wikipedia Mathematics Reference Desk, n^4 Divides Central Binomial Coefficient.

Example

The central binomial coefficient C(2*154836,154836) is divisible by 154836^3.

Maple

A282163 := proc (n, m) local a, cbc, k; a := {}; cbc := binomial(2*n, n); for k from n+1 to m do cbc := cbc*(4-2/k); if type(cbc/k^3, integer) then a := `union`(a, {k}) end if end do; a end proc; A282163(0, 10^6)

Mathematica

Select[Table[n, {n, 10^6}], IntegerQ[Binomial[2#, #]/#^3] &] (* for small n *)
n := 0; m := 10^6; A282163 := {}; cbc := Binomial[2n, n]; For[k := n+1, k <= m, k++, {cbc *= 4-2/k, If[IntegerQ[cbc/k^3], A282163 = Append[A282163, k]]}] (* for large m *)
A282163:={}; k:=3; For[n:=1, n<=10^6, n++, {f=FactorInteger[n], For[j:=1, j<=Length[f], j++, {b=True, If[Sum[Floor[2n/f[[j, 1]]^i]-2 Floor[n/f[[j, 1]]^i], {i, 1, Length[IntegerDigits[2n, f[[j, 1]]]]}]<f[[j, 2]]k, {b=False, Break[]}]}], If[b, A282163=Append[A282163, n]]}] (* Legendre's formula for drastic time reduction, Lucian Craciun, Feb 28 2017; optimized by Lucian Craciun, Mar 02 2017 *)

Crossrefs

Cf. A000108, A000984, A014847, A121943, A282346, A283073, A283074, A282672.

Sequence in context: A256951 A101767 A235220 * A250883 A233816 A050520

Adjacent sequences: A282160 A282161 A282162 * A282164 A282165 A282166

Keyword

nonn

Author

Lucian Craciun, Feb 07 2017

Status

approved