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
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
KEYWORD
nonn
AUTHOR
Lucian Craciun, Feb 07 2017
STATUS
approved